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# Differentiation Rules

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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.

## Chain rule

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

## Product rule

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 :

## Quotient rule

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 :

## Differentiation Rules - Key takeaways

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

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

To differentiate using the quotient rule you use the following formula when y = u/v, dy/dx = (v du/dx - u dv/dx)/v^2

## Final Differentiation Rules 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

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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