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# Differentiation

Differentiation is a method of finding rates of change, i.e. the gradients of functions. The result of differentiating a function is called the derivative of that function.

## The process of differentiation

The process of differentiation is represented by . This is equivalent to 'change in y divided by change in x'. Variables x and y can be substituted for any other letter.

Some alternative notation for derivatives involves an apostrophe '. Differentiating a function y 'with respect to x' (meaning x is the value on the bottom of the fraction) results in the derivative y '. If the function is represented as f (x), then its derivative can be represented as f '(x).

Let's do a quick review of how to find the gradient of a straight line graph:

Example of a gradient of a straight line graph, Heale - StudySmarter Originals

However, if we look at a quadratic graph, it isn't clear how to find its gradient. This is because it changes at different points in the graph as the line curves, getting more or less steep.

One potential method we could use is to draw a tangent at a given point and find its equation. However, this would only give us the gradient at that point - what if we wanted to find a general expression for the gradient of any point on the graph?

We use differentiation to find a function for the gradient of a graph. The method is very straightforward - you need to:

• Decrease the power of x by one

• Multiply by the old power

Therefore, as a general rule, when differentiating , your result is .

## How do you differentiate a polynomial?

Let's say we have the following graph of and we want to find the gradient at the point .

To differentiate the function, we take each power of x and perform the above steps on it - reduce the power by 1, and multiply by the old power.

2 isn't a power of x, so we can't apply our usual method here.

To understand how to differentiate it, we need to look at the representation of differentiation . As a reminder, this means 'the change in y divided by the change in x'.

Since 2 is a constant, changes in x and y do not affect its value, and vice versa. This effectively means that for the gradient it doesn't matter what the value is - it is only important in the context of the original function. For this reason, the derivative of a constant is defined as 0.

Now that we have found the derivative of each of the terms in our function, we can create a function for the gradient at any given point:

Therefore to find the gradient at the point where , substitute this value into our new equation:

## What is differentiation from first principles?

Differentiation from first principles tells us about the concept of differentiation.

Let's consider this curve which is part of a graph that we would like to differentiate. We have chosen two points along it, (x, f (x)) and (x + h, f (x + h)), and we would like to find the gradient at the point (x, f (x)):

We know to find the gradient between these points, we find the change in y divided by the change in x:

The closer we move those two points together, the better our estimate of the gradient at (x, f (x)) will be. As h gets closer and closer to 0, the estimate will be better and better. We can write this as the formula:

We know the derivative of is , but we can prove this by substituting it into the formula:

Finally, we need to consider what happens at the limit as h approaches 0: h disappears, and we are just left with our answer 2x.

## What can differentiation tell us about graphs?

Differentiation can tell us a lot about the nature of graphs and their turning points. These are also known as critical points as they are points where the gradient is equal to zero. There are three possibilities when this is the case:

When the graph is quadratic, it's obvious if the critical point is a maximum or minimum, as there is only one, and all you need to do is consider the shape of the graph (using the coefficient of the term). However, when there are multiple critical points, it isn't so clear.

In order to determine the nature of a critical point for cubic graphs, you need to check the gradients on either side of it.

Let's consider a local maximum:

We can see that the first part of the graph is increasing according to the direction of the graph, then after the critical point, it starts to decrease.

If we found the gradient of the increasing part of the graph, it would be positive, and the decreasing part would be negative. In summary:

increasing

critical point

decreasing

Let's look at determining the nature of a critical point.

We already know that the critical point of this graph is going to be a minimum, because the has a positive coefficient. However, we'll prove it using differentiation.

First, we need to differentiate the function;

Now we need to find the coordinates of the critical point, the x value where the derivative of the function is zero. We can do this by solving the equation , since we know the gradient is zero at that point.

Now we can create a simple table and sub in the values of x on either side:

Since the gradient on the left is decreasing and the gradient on the right is increasing, we have shown that the turning point is a minimum.

If the gradient on the left would be increasing and the gradient on the right decreasing, the turning point would be a maximum.

Finally, if they are both increasing or both decreasing, it must be a stationary point.

### What can the second derivative tell us about graphs?

A different possibility to determine if a critical point is a maximum, minimum, or stationary point is by using the second derivative, as the second derivative of a graph tells you its curvature.

• A positive curvature means the graph curves towards the left if considered along the x-axis (minimum).

• A negative curvature means that the graph curves towards the right (maximum).

• If the second derivative of a function is zero at a certain point, the curvature is zero, and the graph is straight at this point (stationary point).

In our example:

This means that the curvature is positive anywhere on the graph and the critical point is a maximum.

## What are some rules of differentiation?

Some differentiation rules which help you to find the derivatives of more complex functions are:

### The product rule

The product rule can be used to find the derivative of two functions multiplied together. The formula is;

If y = uv

Where u is the function f(x) and v is the function g(x), and f'(x), g'(x) are their derivatives u' and v'.

Differentiate the function

We could expand the brackets in this example and find the derivative the usual way, however often using the product rule is faster and less prone to error.

To use the product rule on this function, we need to let and .

Next, we need to differentiate them individually:

Finally, we substitute these values into the product formula:

### The quotient rule

The quotient rule can be used to find the derivative of two functions divided by each other. The formula is:

Where u is the function f (x) and v is the function g (x), and f '(x), g' (x) are their derivatives u' and v'.

Differentiate the function

We let u be the numerator, and v be the denominator, ie and , then differentiate them individually as before to get and .

Finally, we need to substitute these values into the formula:

### The chain rule

The chain rule can be used to find the derivative of a function of a function. The formula is;

Differentiate the function

We let , then substitute this into the main equation such that . We then differentiate them both individually, thus finding and ;

Finally, we multiply them together to get , and substitute u back in to get .

## Parametric differentiation

Sometimes we want to differentiate functions where x and y are both in terms of a third variable. In these situations, we need to use parametric differentiation.

We can use the chain rule to differentiate in terms of x and y:

We could rearrange the equation involving x to be in terms of t. The above equation could also be written as the following, making it easier to differentiate:

Let's first try rearranging and multiplying our results:

Now let's try the second method to ensure we get the same answer. All we need to do is differentiate each equation individually with respect to t, and then divide by :

## Implicit differentiation

When differentiating, we are usually faced with explicit functions - that is, functions which generally look like However, what if we wanted to differentiate the equation ?

We need to use a technique called implicit differentiation to solve this. We can approach each part of the equation separately and write:

We know how to differentiate two of the parts. The first stage to differentiating the y part is to differentiate it as normal, but leave ;

Now we need to rearrange the equation in terms of :

## Differentiation - key takeaways

• Differentiation is a method of finding rates of change, i.e. gradients of functions.

• The result of a differentiation calculation is called the derivative of a function.

• The process of differentiation is represented by .

• To differentiate a polynomial:
• Decrease the power of x by one

• Multiply by the old power

• The derivative of a constant is defined as 0.
• Differentiation from first principles uses the formula,
• increasing
• critical point
• When the derivative is equal to zero, there are three possibilities:

• decreasing

• The product rule is

• The quotient rule is

• The chain rule is

• Parametric differentiation uses the formula

• Implicit differentiation involves differentiating each part of the equation separately and rearranging for

To differentiate a fraction, you need to use the quotient rule;

y'=(vu'-uv')/v^2

Differentiation is the process of finding a function for the gradient of a given function.

To differentiate a function of a function, you need to use the chain rule; dy/dx=dy/du ⋅ du/dx

## Final Differentiation Quiz

Question

What are the three differentiation rules you need to know?

Chain rule, product rule, quotient rule.

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Question

When should you use the chain rule?

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

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Question

When should you use the product rule?

This rule can be used when you are differentiating the product of two functions.

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Question

When should you use the quotient rule?

This rule is used when you are differentiating a function that is being divided by another function, otherwise known as a quotient function.

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Question

What is the product rule?

The product rule is a rule used for differentiation.

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Question

When do you use the product rule?

You can use the product rule when you are differentiating the products of two functions.

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Question

What is the chain rule?

The chain rule is a rule used in differentiating functions.

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Question

When should you use the chain rule?

The chain rule can be used when differentiating a composite function, also known as a function of a function.

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Question

Differentiate from first principles 3x.

3

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Question

Differentiate from first principles 5x.

5

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Question

What is the quotient rule?

The quotient rule is a rule used in differentiation, it is used when you are differentiating a quotient.

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Question

What is a quotient function?

A quotient function is a function that is being divided by another function.

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Question

What are parametric equations?

Parametric equations are two equations dependent on a common third variable.

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Question

What is the difference between differentiating Cartesian equations and differentiating parametric equations?

In Cartesian form the chain rule is applied, and the derivatives of two functions are multiplied but in parametric differentiation the reverse chain rule is applied and the two derivatives are divided .

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Question

How do we find the slope of the tangent of a parametric curve?

By finding the derivative of the curve.

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Question

How do we find the slope of a parametric curve?

Using parametric differentiation.

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Question

What is the normal of a curve?

It is the straight line perpendicular to the tangent of a curve.

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Question

What is the tangent of a curve?

It is a straight line that touches a curve at a single point.

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Question

Is parametric derivative equal to the slope of the curve?

No

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Question

Is the parametric derivative equal to the slope of the tangent of a curve?

Yes

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Question

Is parametric derivative equal to the slope of a normal to a curve?

No

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Question

Can parametric differentiation be used to construct equations of normals and tangents?

Yes.

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Question

What is the reverse chain rule?

Is the rule used for parametric differentiation.

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Question

Is the parametric derivative in terms of the third common parameter?

No.

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Question

Do we differentiate all of the variables for implicit differentiation ?

Yes, each term is differentiated as normal on both sides, including unknown variables x,y.

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Question

What happens to the y term?

The y term also needs to be multiplied by dy/dx and then is isolated to one side.

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Question

What happens to the y term?

The y term also needs to be multiplied by dy/dx and then is isolated to one side.

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Question

What is higher order implicit differentiation?

Higher order implicit differentiation is differentiation of the implicit derivative .

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Question

How do we find the slope of the tangent of a curve?

By performing implicit differentiation.

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Question

What is the slope of a normal of a curve?

It is the negative reciprocal of the slope of the curve.

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Question

Is the implicit derivative the slope of the normal of a curve?

No

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Question

What is the difference between implicit and explicit differentiation?

In implicit differentiation the y term is also multiplied by dy/dx.

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Question

What is the difference between parametric differentiation and implicit differentiation?

In parametric differentiation there is not a third parameter.

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Question

Is implicit differentiation equal to the slope of the tangent of a curve?

Yes

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Question

Do all of the differentiation rules apply to implicit differentiation?

Yes

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Question

Why is implicit differentiation used?

We can find  dy/dx without the need of solving for y.

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Question

When is implicit differentiation used?

When implicit functions are present.

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