StudySmarter - The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

Combining Differentiation Rules

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

If you are reading this article, you should have studied the basic Differentiation Rules and delved into the more complex rules like The Power Rule, The Product Rule, The Quotient Rule, and The Chain Rule.

Now you are ready to put together all you have learned so far!

Most functions you deal with in calculus are more complicated than what you have learned so far, and, of course, you will be asked to find the derivatives of these more complicated functions.

How can you do this?

By combining differentiation rules!

These more complicated functions that you deal with in calculus are just made up of simpler functions that have been combined in one (or more) of the following ways:

Addition and subtraction: \( f(x)+g(x) \) and \( f(x)-g(x) \)

Multiplication and division: \( f(x) g(x) \) and \( \frac{f(x)}{g(x)} \)

Function composition: \( f(g(x)) \)

Look familiar? These combinations of functions have their own differentiation rules!

- For the addition and subtraction of functions, you use the sum and difference rules.
- For the multiplication and division of functions, you use the product and quotient rules.
- For the composition of functions, you use the chain rule.

Let's quickly recap these differentiation rules:

Say you have two differentiable functions, \( f(x) \) and \( g(x) \). For these, the following derivative rules apply:

- Sum Rule: \( \left( f(x)+g(x) \right)' = f'(x)+g'(x) \)
- Difference Rule: \( \left( f(x)-g(x) \right)' = f'(x)-g'(x) \)
- Product Rule: \( \left( f(x) g(x) \right)' = f'(x)g(x)+f(x)g'(x) \)
- Quotient Rule: for \( g(x) \neq 0, \left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x)-f(x)g'(x)}{ \left( g(x) \right)^{2} } \)
- Chain Rule: \( \left( f(g(x)) \right)' = f'(g(x))g'(x) \)

You know that these rules can be used one at a time. But you can also use them together. This means you can differentiate any combination of elementary functions (provided they are differentiable, of course).

Do not be fooled; however, just because you can use these rules together does not mean the process is easy. Sure, it is definitely easier than finding the derivative using the first principle^{1}, but this is no trivial exercise. Are you up for this challenge?

There are **several things to consider** when combining differentiation rules to find the derivative of a function:

- Identify which differentiation rules to use.
- Determine in which order to apply those rules.
- Ask yourself: “Are there any algebraic simplifications I can make to turn this process easier?”

But before you start using multiple differentiation rules together, let's work out a **strategy** for doing so. Start by implementing the rule of thumb:

**Rule of thumb**:

Apply the differentiation rules in the **reverse order** in which we would want to evaluate the function.

What exactly does this mean, though?

This means you find the derivative of your function by **working from the outside in**, by breaking down the complicated function into smaller parts.

**Strategy for combining differentiation rules**

Let's say you have the function:

\[ f(x) = \left( \frac{x^{2}+4}{x^{3}-3x+6} \right)^{4} + \sqrt{2x-5} \]

How can you find its derivative?

**Strategy**:

1. Break the overall function into parts, working from the outside in. In this case, the outermost layer is where the main function \( f(x) \) is a sum of two functions.

\[\begin{align}f(x) & = \underbrace{ \left( \frac{x^{2}+4}{x^{3}-3x+6} \right)^{4} }_{g(x)} + \underbrace{ \sqrt{2x-5} }_{h(x)} \\& = g(x) + h(x)\end{align}\]

Based on the **sum rule** of differentiation, you know you can differentiate \( g(x) \) and \( h(x) \) separately and add them together after.

2. Now, if you look at \( g(x) \), the outermost layer of this function is something to the power of \( 4 \). You can write this as:

\[\begin{align}g(x) & = \left[ \underbrace{ \left( \frac{x^{2}+4}{x^{3}-3x+6} \right) }_{u(x)} \right]^{4} \\& = \left( u(x) \right)^{4}\end{align}\]

This simplification shows you that \( g(x) \) is a composition of functions. Remember how to differentiate a composition of functions?

That's right, the **chain rule**!

3. Following the pattern you have started in the previous two steps, you want to continue to **remove layers of complexity** from the original function, \( f(x) \), until all that is left are the elementary functions you know how to differentiate. The breakdown below shows you how you can do this:

If you look at the five expressions at the bottom of the tree:

- \( x^{2} + 4 \);
- \( x^{3} - 3x + 6 \);
- \( u^{4} \);
- \( \sqrt{v} \); and
- \( 2x - 5 \),

you can see that all of these are expressions that you know how to differentiate using one of the seven differentiation rules.

If you apply the correct differentiation rules to each expression at each stage of the breakdown, you see that you can find the derivative of even the most complicated functions.

Now that you have devised a strategy for combining differentiation rules let's walk through a simple example.

**Find the derivative of a polynomial using the sum, constant multiple, power, and product rules**.

Given the function:

\[ f(x) = 4g(x)+x^{3}h(x) \]

Find \( f'(x) \).

**Solution**:

1. The first step to any differentiation problem is to **analyze the given function** and determine which rules you want to apply to find the derivative.

- Looking at the outermost layer of complexity, you see that \( f(x) \) is a sum of two functions. So, you need to use the
**sum rule**. - When you look at these two functions separately, you see that the first one, \( 4g(x) \), is a constant multiplied by a function, and the second, \( x^{3}h(x) \), is a product of two functions. So, to differentiate these, you need to use the
**constant multiple rule**for the first function and the**product rule**for the second. - Lastly, you see that to differentiate the \( x^{3} \) in the second function, you must use
**the power rule**.

2. Starting with the outermost layer of complexity, **apply the sum rule**.

\[ f'(x) = \frac{d}{dx} \left( 4g(x)+x^{3}h(x) \right) = \frac{d}{dx} \left( 4g(x) \right) + \frac{d}{dx} \left( x^{3}h(x) \right) \]

3. Moving to the next layer of complexity, **apply the constant multiple rule **to differentiate \(4g(x)\) and the **product rule** to differentiate \(x^{3}h(x)\).

\[ f'(x) = 4 \frac{d}{dx} (g(x)) + \left( \frac{d}{dx} \left( x^{3} \right) \cdot h(x) + \frac{d}{dx} \left( h(x) \right) \cdot x^{3} \right) \]

4. Finally, **take the derivatives** (using the power rule for \( x^{3} \)) and **simplify**.

\[ \bf{ f'(x) } = \bf{ 4g'(x) + 3x^{2}h(x) + h'(x)x^{3} } \]

Now, let's move on to a common occurrence in differential calculus: finding the derivative of a function using both the product and the quotient rules.

**Combining the product and quotient rules** (and a few others).

Given the function:

\[ f(x) = \frac{5x^{2}g(x)}{3x+2} \]

Find \( f'(x) \).

**Solution**:

1. Again, the first step is to **analyze the given function** and determine which rules you want to apply (and the best order to apply them) to find the derivative.

- Since this is a rational function, you know you will need to use the
**quotient rule**. And, since the outermost layer of complexity of \( f(x) \) is the division of two functions, you should apply the quotient rule first. - If you look at the function in the numerator, you can see that it is a product of two functions, so you need the
**product rule**here.- Breaking down the function in the numerator, you see you need the
**power rule**to find the derivative of \( 5x^{2} \).

- Breaking down the function in the numerator, you see you need the
- If you look at the function in the denominator, you can see that its derivative is simple to find using the
**constant multiple and constant rules**.

2. Starting with the outermost layer of complexity, **apply the quotient rule**.

\[ f'(x) = \frac{ \frac{d}{dx} \left( 5x^{2}g(x) \right) (3x+2) - \frac{d}{dx} (3x+2) \left( 5x^{2}g(x) \right)}{\left( 3x+2 \right)^{2}} \]

3. Now you can **apply the product rule** to find \( \frac{d}{dx} \left( 5x^{2}g(x) \right) \). At the same time, you can **apply the constant multiple and constant rules** to find the derivative: \( \frac{d}{dx} (3x+2) = 3 \).

\[ f'(x) = \frac{ \left( \frac{d}{dx} \left( 5x^{2} \right)g(x) + g'(x) \left( 5x^{2} \right) \right)(3x+2) - 3 \left( 5x^{2}g(x) \right) }{\left( 3x+2 \right)^{2}} \]

4. From here, you can apply the **power rule** to find the derivative of \( 5x^{2} \).

\[ f'(x) = \frac{ \left( 10x g(x) + g'(x) \left( 5x^{2} \right) \right)(3x+2) - 3 \left( 5x^{2}g(x) \right) }{\left( 3x+2 \right)^{2}} \]

Now, you can either stop here, as you have found the derivative, or you can expand and simplify the equation. The simplified form of this derivative is:

\[ \bf{ f'(x) } = \bf{ \frac{15x^{3}g'(x)+15x^{2}g(x)+10x^{2}g'(x)+20xg(x)}{ \left( 3x+2 \right)^{2}} } \]

When taking the derivative of more complex functions, sometimes the answer doesn't get as simple as you'd like!

Next is learning how to take the derivative of a combination of functions. This is where **the chain rule** comes into play. And, since the chain rule is often used with the power rule, there is a particular case for the power rule of a composition of functions that combines the power and chain rules:

**Rule: the Power Rule for a composition of functions**

For all values of \( x \) that the derivative of \( f(x) \) is defined, if:

\[ f(x) = \left( g(x) \right)^{n}, \]

then:

\[ f'(x) = n \left( g(x) \right)^{n-1} g'(x). \]

**Combining the chain and power rules**.

What is the derivative of the following function?

\[ f(x) = \frac{1}{ \left( 3x^{2}+1 \right)^{2} } \]

**Solution**:

1. Before you start using the derivative rules here, there is an algebraic simplification you can use to make using the chain rule easier. Rewrite the function as:

\[ f(x) = \left( 3x^{2}+1 \right)^{-2} \]

2. Break the function into its elementary parts:

\[ \begin{align}f(x) &= \left( \underbrace{ \left( 3x^{2}+1 \right) }_{g(x)} \right)^{-2} \\&= \left( g(x) \right)^{-2}\end{align} \]

3. Now, you have \( f(x) \) in the same form as **the power rule for a composition of functions**. So, the next step is to work from the outside in, first applying the power rule for a composition of functions and then the power rule on \( 3x^{2}+1 \) to find the derivative.

\[ \begin{align}f'(x) &= n \left( g(x) \right)^{n-1} g'(x) \\&= -2 \left( 3x^{2}+1 \right)^{-2-1} \frac{d}{dx} \left( 3x^{2} +1 \right) \\&= -2 \left( 3x^{2}+1 \right)^{-3} (6x)\end{align} \]

4. It is bad practice to leave negative exponents, so the final step is to rewrite the derivative of the function without negative exponents:

\[ \bf{ f'(x) } = \bf{ \frac{-12x}{ \left( 3x^{2}+1 \right)^{3} } } \]

**Combining the chain and power rules with a trig function**.

What is the derivative of the following function?

\[ f(x) = sin^{3}(x) \]

**Solution**:

1. The first step here is to remember that \( sin^{3}(x) = (sin(x))^{3} \). Rewrite the function as:

\[ f(x) = (sin(x))^{3} \]

2. From here, you can see that this is of the form \( f(x) = \left( g(x) \right)^{n} \), so you can apply the power rule for a composition of functions here to find the derivative.

\[ \begin{align}f'(x) &= n \left( g(x) \right)^{n-1} g'(x) \\&= 3 (sin(x))^{3-1} \frac{d}{dx}sin(x) \\&= 3 (sin(x))^{2} cos(x) \\\bf{ f'(x) } &= \bf{ 3sin^{2}(x) cos(x) }\end{align}\]

Moving on from chain and power rule combinations, let's investigate how combining the chain rule with other differentiation rules works.

**Combining the chain rule with a general cosine function**.

What is the derivative of the following function?

\[ h(x) = cos \left(g(x) \right) \]

**Solution**:

1. In this case, it is first helpful to think of \( h(x) = cos \left(g(x) \right) \) as \( h(x) = f(g(x)) \). In doing this, you have:

\[ f(x) = cos(x) \]

2. What is the derivative of \( cos(x) \)? It is \( -sin(x) \)! Using this, now you have:

\[ f'(g(x)) = -sin(g(x)) \]

3. Now, you can apply the chain rule.

\[ h'(x) = f'(g(x))g'(x) \]

4. Finally, substitute \( f'(g(x)) = -sin(g(x)) \).

\[ \bf{ h'(x) } = \bf{ -sin(g(x))g'(x) } \]

**The chain rule with a cosine function**.

Using the rule, you derived in the example above, what is the derivative of the following function?

\[ h(x) = cos \left( 5x^{2} \right) \]

**Solution**:

1. Following the previous example, think of \( 5x^{2} \) as \( g(x) \).

\[ \mbox{ if } g(x) = 5x^{2}, \mbox{ then } g'(x) = 10x \]

2. Now, using the result from the previous example:

\[ \begin{align}h'(x) &= -sin(g(x))g'(x) \\&= -sin \left( 5x^{2} \right) \cdot 10x \\\bf{ h'(x) } &= \bf{ (-10x)sin \left( 5x^{2} \right) }\end{align} \]

Now that you can combine the chain rule with the other differentiation rules let's look at combining the chain rule with itself. That is, you can apply the chain rule more than once to find the derivative of a composition of 3 (or more) functions.

**Rule: the chain rule for a composition of 3 functions**.

For all values of \( x \) where the function is differentiable, if

\[ k(x) = h(f(g(x))), \]

then,

\[ k'(x) = h'(f(g(x))f'(g(x))g'(x). \]

While this rule could be helpful, **you do not need to memorize this rule**, as you will achieve the same result when you apply the chain rule multiple times.

**Developing the chain rule for a composition of 3 functions**.

Looking at general functions, you can develop the rule above.

1. Let

\[ k(x) = h(f(g(x))). \]

2. Apply the chain rule once.

\[ \begin{align}k'(x) &= \frac{d}{dx}(h(f(g(x)))) \\&= h'(f(g(x))) \cdot \frac{d}{dx}(f(g(x)))\end{align} \]

3. Apply the chain rule again.

\[ k'(x) = \underbrace{h'(f(g(x)))}_{1} \underbrace{f'(g(x))}_{2} \underbrace{g'(x)}_{3} \]

Note: the derivative of the composition of 3 functions has three parts. This pattern holds for four functions, five functions, and so on.

**Using the differentiation rules to find the derivative of a composite of 3 functions**.

What is the derivative of the following function?

\[ k(x) = cos^{4} \left( 7x^{2} + 1 \right) \]

**Solution**:

1. Rewrite \( k(x) \) to make it easier to work with.

\[ k(x) = \left( cos \left( 7x^{2} + 1 \right) \right)^{4} \]

2. Apply the chain rule several times in a row until the derivative is found.

\[ \begin{align}k'(x) &= 4 \left( cos \left( 7x^{2} + 1 \right) \right)^{3} \left( \frac{d}{dx} \left( cos \left( 7x^{2} + 1 \right) \right) \right) \\&= 4 \left( cos \left( 7x^{2} + 1 \right) \right)^{3} \left( -sin \left( 7x^{2} + 1 \right) \right) \left( \frac{d}{dx} \left( 7x^{2} + 1 \right) \right) \\&= 4 \left( cos \left( 7x^{2} + 1 \right) \right)^{3} \left( -sin \left( 7x^{2} + 1 \right) \right) (14x) \\\end{align} \]

3. Simplify the answer.

\[ \bf{ k'(x) } = \bf{ -56x \, sin \left( 7x^{2} + 1 \right) cos^{3} \left( 7x^{2} + 1 \right) } \]

**Using the differentiation rules to find the derivative of a polynomial function at a point**.

What is the derivative of the following function at the point \( (1, -4) \)?

\[ f(x) = (x-5)(x-2)^{6} \]

**Solution**:

1. Think about the course of action you want to take.

- Since this is a factored polynomial, you could expand and simplify the polynomial, then take the derivatives of each component, but would that be the most efficient method?
- The short answer is no, it's not.
- Instead, it is more efficient (i.e., faster and easier) to view \( f(x) \) as a product of two functions:

\[ \begin{align}f(x) &= \underbrace{(x-5)}_{g(x)} \underbrace{(x-2)^{6}}_{h(x)} \\&= g(x)h(x)\end{align} \]

2. To use the product rule to find this derivative, you first need to know what \( g'(x) \) and \( h'(x) \) are.

- The first function, \( g(x) = x-5 \), is an elementary function. You can differentiate this function using the power rule to get:

\[ g'(x) = 1 \]

- The second function, \( h(x) = (x-2)^{6} \), is a composition of functions. You can break it down like this:

\[ \begin{align}h(x) &= {\underbrace{(x-2)}_{v(x)} }^{6}\\&= (v(x))^{6}\end{align} \]

- So, if you let \( u(x) = x^{6} \) and \( v(x) = x-2 \), then \( h(x) = u(v(x)) \). You can differentiate this using the chain rule.
- But, to use the chain rule here, you first need to find \( u'(x) \) and \( v'(x) \).
- Using the power rule, \( u'(x) = 6x^{5} \).
- Using the constant multiple and constant rules, \( v'(x) = 1 \).

- Substituting \( u'(x), v'(x) \), and \( v(x) \) into the chain rule to solve for \( h'(x) \), you get:

- But, to use the chain rule here, you first need to find \( u'(x) \) and \( v'(x) \).

\[ \begin{align}h'(x) &= u'(x-2) \cdot 1 \\&= 6(x-2)^{5}\end{align}\]

3. With \( g'(x) \) and \( h'(x) \) found, you can substitute the following into the product rule:

- \( g(x) = x-5 \)
- \( h(x) = (x-2)^{6} \)
- \( g'(x) = 1 \)
- \( h'(x) = 6(x-2)^{5} \)

4. Once substituted, you get:

\[ \begin{align}f'(x) &= g'(x)h(x)+g(x)h'(x) \\&= 1 \cdot (x-2)^{6} + (x-5) \cdot 6(x-2)^{5} \\&= (x-2)^{5} \left( (x-2) + 6(x-5) \right) \\&= (x-2)^{5} (7x-32)\end{align} \]

5. Now that you have the derivative of the function, all you need to do is evaluate the derivative at the point \( (1, -4) \). To do this, you substitute the x-coordinate of the point into the function's derivative and solve.

\[ \begin{align}\left. f'(x) \right|_{x=1} &= (1-2)^{5} (7 \cdot 1 - 32) \\&= (-1)^{5} (-25) \\&= 25\end{align}\]

6. Therefore,

\[ \bf{ \left. f'(x) \right|_{x=1} } = \bf{ 25 } \]

**Using the differentiation rules to find the derivative of a rational function at a point**.

What is the derivative of the following function at point \( (1, -1) \)?

\[ y = \frac{-2x}{\sqrt{3x^{2}+1}} \]

**Solution**:

1. Think about the course of action you want to take.

- What is the most efficient method to find the derivative?
- In this case, the outermost layer of complexity is the fact that the given function is a quotient of two functions.

\[ \begin{align}y &= \frac{ \overbrace{-2x}^{u(x)} }{ \underbrace{\sqrt{3x^{2}+1}}_{v(x)} } \\&= \frac{u(x)}{v(x)}\end{align}\]

2. The first thing you will want to do is use the quotient rule, where \( u(x) = -2x \) and \( v(x) = \sqrt{3x^{2}+1} \).

- But, to use the quotient rule, you need to find \( u'(x) \) and \( v'(x) \).
- Using the constant multiple rule, \( u'(x) = -2 \).
- \( v(x) \) is a composition of functions: \( f(x) = \sqrt{x} \) and \( g(x) = 3x^{2}+1 \), so you will need to use the chain rule to find \( v'(x) \).
- Using the power rule, \( f'(x) = \frac{1}{2\sqrt{x}} \) and \( g'(x) = 6x \).

- Substituting \( f'(x), g'(x), g(x) \) into the chain rule gives you:

\[ v'(x) = \frac{3x}{\sqrt{3x^{2}+1}} \]

3. Substitute the following into the quotient rule:

- \( u(x) = -2x \)
- \( v(x) = \sqrt{3x^{2}+1} \)
- \( u'(x) = -2 \)
- \( v'(x) = \frac{3x}{\sqrt{3x^{2}+1}} \)

\[ \begin{align}y' &= \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^{2}} \\&= \frac{-2 \sqrt{3x^{2}+1}+2x \frac{3x}{\sqrt{3x^{2}+1}}}{3x^{2}+1}\end{align} \]

4. Multiply the numerator and denominator by \( \sqrt{3x^{2}+1} \) to simplify this fraction:

\[ \begin{align}y' &= \frac{-2 \sqrt{3x^{2}+1}+2x \frac{3x}{\sqrt{3x^{2}+1}}}{3x^{2}+1} \cdot \frac{\sqrt{3x^{2}+1}}{\sqrt{3x^{2}+1}} \\&= \frac{-2 \left( 3x^{2}+1 \right) + 2x \cdot 3x}{(3x^2 + 1) \sqrt{3x^{2} + 1}} \\&= \frac{-6x^{2}-2+6x^{2}}{(3x^2 + 1) \sqrt{3x^{2} + 1}} \\&= \frac{-2}{(3x^2 + 1) \sqrt{3x^{2} + 1}}\end{align} \]

5. Now that you have the derivative of the function, all you need to do is evaluate the derivative at the point \( (1, -1) \). To do this, you substitute the x-coordinate of the point into the function's derivative and solve.

\[ \begin{align}\left. y'(x) \right|_{x=1} &= \frac{-2}{(3(1)^2 + 1) \sqrt{3(1)^{2} + 1}} \\&= \frac{-2}{(3 + 1) \sqrt{3 + 1}} \\&= \frac{-1}{4}\end{align}\]

6. Therefore,

\[ \bf{ \left. y'(x) \right|_{x=1} } = \bf{ \frac{-1}{4} } \]

- We can calculate the derivatives of any combination of elementary functions using the s differentiation rules:
- Constant Rule
- Constant Multiple Rule
- Power Rule
- Sum & Difference Rules
- Product Rule
- Quotient Rule
- Chain Rule

- Thinking about the order in which to apply the differentiation rules will help us ensure we choose the easiest or most efficient method.
- In general, we want to work from the outside in of the function we wish to differentiate. This helps us decompose the function into parts that can be differentiated easily.
- Keep in mind that some functions can be simplified so that we do not need to use multiple differentiation rules to find its derivative.

- https://www.onlinemath4all.com/how-to-find-derivatives-using-first-principle.html

- Constant Rule
- Constant Multiple Rule
- Power Rule
- Sum & Difference Rules
- Product Rule
- Quotient Rule
- Chain Rule

We can find the derivative of a function using the 7 differentiation rules by:

- Identifying which of the 7 differentiation rules to apply.
- Determining in which order to apply those rules by following the strategy of working from the outside in.

More about Combining Differentiation Rules

60%

of the users don't pass the Combining Differentiation Rules quiz! Will you pass the quiz?

Start QuizBe perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.