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Differentiation Rules

- Calculus
- Absolute Maxima and Minima
- Accumulation Function
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- Algebraic Functions
- Alternating Series
- 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
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- 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 Sin, Cos and Tan
- Determining Volumes by Slicing
- Disk Method
- Divergence Test
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Improper Integrals
- Initial Value Problem Differential Equations
- Integral Test
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- Integration Tables
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- Net Change Theorem
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- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Tangent Lines
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
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- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Washer Method
- Decision Maths
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- Angles in Circles
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- Area and Volume
- Area of Circles
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- Algebra
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- Area and Perimeter of Quadrilaterals
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- Differentiation
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- Differentiation of Hyperbolic Functions
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- Rounding
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- SSS Theorem
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- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- 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
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- Statistics
- Binomial Distribution
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- Central Limit Theorem
- Comparing Data
- Conditional Probability
- Correlation Math
- Cumulative Frequency
- Data Interpretation
- Discrete Random Variable
- Distributions
- Events (Probability)
- Frequency Polygons
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Testing
- Large Data Set
- Linear Interpolation
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Random Variables
- Sampling
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Measures
- Tree Diagram
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

There are many different rules that can be used when differentiating, and each rule can be used for a specific reason. There are three different rules that you will need to know:

Chain rule

Product rule

Quotient rule

It is important to note that not all of the below formulae are written in the provided formula booklet and you will have to memorise them for your exam.

The chain rule can be used when you are differentiating a composite function, which is also known as a function of a function. The formula for this rule is below. This is when y is a function of u and u is a function of x:

Using the chain rule, differentiate

First, you can start by rewriting it in terms of y and u:

Now you can find the first part of your chain rule formula , by differentiating your y:

Next, you can find the second part of your chain rule formula , by differentiating your u:

Now that you have found both parts of your sum you can multiply them together to find :

Finally, it is important to express your answer in terms of x, and to do this you can use :

The function that you are given may involve a trigonometric function. Let's work through an example to see how this would be solved.

If find

Let’s start by identifying your u and y:

Now you can look at the chain rule formula, break it down, and find each part:

Next, you can put your and into the formula to find :

Finally, you need to make sure your answer is written in terms of x, and you can do this by substituting in :

The chain rule can also be written in notation form, which allows you to differentiate a function of a function:

If then

The product rule is used when you are differentiating the product of two functions. A product of a function can be defined as two functions being multiplied together. When using this rule you need to make sure you have the product of two functions and not a function of a function, as they can be confused. The formula for this rule is below – this is if y=uv when u and v are functions of x:

This function can also be written in function notation:

If then

Given that find

Looking at the formula, first, you need to identify each part of the formula:

To find and you need to differentiate your u and v:

Now you have found each part of the formula you can solve for :

Given that find

To do this you can start by looking at the formula and identifying what you need:

Now you can differentiate your u and v to find the next part of the formula:

Input all of the information into the formula to find :

The quotient rule is used when you are differentiating the quotient of two functions, this is when one function is being divided by another. The formula used for this rule is below, this is when :

This can also be written in function notation:

If then

If find :

First, you can start by looking at your formula and find each part of it:

Now you can solve the formula with all of the above information:

Let’s look at a second example involving a trigonometric function.

If find

Just like in the previous example, you can start by looking at your formula and finding each part of it to find your :

Now that you have each part of the formula you can substitute them into the formula and find :

There are three main differentiation rules, chain rule, product rule, and quotient rule.

Each rule is used for a different reason and has a different formula for you to use.

The chain rule is used when you are differentiating a composite function.

The product rule is used when you are differentiating the products of two functions.

The quotient rule is used when you are differentiating the quotient of two functions.

To differentiate using the chain rule you use the following formula: dy/dx = dy/du X du/dx

More about Differentiation Rules

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