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# Chain Rule

The chain rule is one of the rules used in differentiationit can be used to differentiate a composite function. A composite function combines two or more functions to create a new function and can also be referred to as a function of a function.

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## Chain rule formula

There is a formula for using the chain rule, when y is a function of u and u is a function of x:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

The formula can also be written in function notation,

if $$y = f(g(x))$$ then$$\frac{dy}{dx} = f'(g(x))g'(x)$$

### Examples using the formula and function notation

Let's look at some examples of the chain rule to help you understand it further:

If $$y = (2x - 1)^3$$ find $$\frac{dy}{dx}$$

First, you can start by looking at the formula for the chain rule before rewriting your y in terms of both y and u:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

$$y = (u)^3$$ $$u = 2x -1$$

Next you can take your y and u and differentiate them to find: $$\frac{dy}{du} \space \frac{du}{dx}$$

$$y = (u)^3$$

$$\frac{dy}{du} = 3u^2$$

Now you can differentiate your u to find : $$\frac{du}{dx}$$

$$u = 2x - 1$$

$$\frac{du}{dx} = 2$$

Now that you have each aspect of the formula you can find $$\frac{dy}{dx}$$:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

$$\frac{dy}{dx} = 3u^2 \cdot 2$$

$$\frac{dy}{dx} = 6u^2$$

Lastly, you need to make sure your answer is written in terms of x, to do this you can substitute in $$u = 2x-1$$:

$$\frac{dy}{dx} = 6u^2$$

$$\frac{dy}{dx} = 6(2x -1)^2$$

The question may also involve some trigonometric functions. Let's look at an example of how to work through it.

If $$y = (\sin x)^5$$ find $$\frac{dy}{dx}$$

You can start this just like before, finding each aspect of your formula:

$$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$$

$$y = (u)^5$$ $$u = \sin x$$

Next you can differentiate both y and u to find $$\frac{dy}{du}$$ and $$\frac{du}{dx}$$:

$$\frac{dy}{du} = 5u^4$$ $$\frac{du}{dx} = \cos x$$

Now that you have all the aspects you can solve to find : $$\frac{dy}{dx}$$

$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$

$\frac{dy}{dx} = 5u^4 \cdot \cos x$

Once again, you need to make sure your answer is written in terms of x. To do this, you have to substitute back in $$u = \sin x$$:$\frac{dy}{dx} = 5u^4 \cdot \cos x$$\frac{dy}{dx} = 5(\sin x)^4 \cdot \cos x$

You may be given the question in function notation form and asked to differentiate.

Differentiate $$f(g(x)) = (3x^2 + 2)^2$$

First, you need to start by looking at your function notation formula:

If $$y = f(g(x))$$ then$$\frac{dy}{dx} = f'(g(x))g'(x)$$

Now you can identify your f(x) and g(x):

$$f(x) = x^2$$ $$g(x) = 3x^2 + 2$$

Next, you can differentiate f(x) and g(x) to find f'(x) and g'(x):

$$f'(x) = 2x \qquad g'(x) = 6x$$

For the formula you also need to find: f'(g(x))

$$f'(g(x)) = 2(3x^2 + 2)$$

Now that you have every aspect of the function notation formula, you can substitute each part in and find $$\frac{dy}{dx}$$:

$$\frac{dy}{dx} = f'(g(x))g'(x)$$

\begin{align} \frac{dy}{dx} &=2(3x^2 + 2)(6x) \\ &= (6x^2 + 4)(6x) \\ &= 36x^3 + 24x \end{align}

### What if the function is not in the form y = f(x)

It is important to consider the formula you would use if the function you are given is not in the form $$y = f(x)$$. The formula to use for this is:

$$\frac{dy}{dx} = \frac{1}{dx/dy}$$

The question could look something like this:

Find the value of $$\frac{dy}{dx}$$ at the point (4, 1) on the curve $$y^4 + 2y = x$$.

Let's work through this question to see how you would solve it. First, you can start by differentiating the equation with respect to y:

$$y^4 + 2y = x$$

$$\frac{dx}{dy} = 4y^3 +2$$

Next, you substitute your differentiated equation into the formula,

$\frac{dy}{dx} = \frac{1}{dx/dy}$

$$\frac{dy}{dx} = \frac{1}{4y^3 + 2}$$

Now all you need to do is substitute the y from the point on the curve from the question into the formula to find your answer:

$\frac{dy}{dx} = \frac{1}{4y^3 + 2}$

$\frac{dy}{dx} = \frac{1}{4(1)^3 + 2}$

$\frac{dy}{dx} = \frac{1}{6}$

Find the value of $$\frac{dy}{dx}$$ at the point (6, 3) on the curve $$4y^2 + 3y = x$$

Once again you start by differentiating the equation with respect to y:

$$4y^2 + 3y = x$$

$$\frac{dx}{dy} = 8y + 3$$

Now you can input that into the formula to find the value of $$\frac{dy}{dx}$$ at the point (6,3): $\frac{dy}{dx} = \frac{1}{dx/dy}$

$\frac{dy}{dx} = \frac{1}{8y+3}$

Next you substitute the y value from the coordinates in order to solve the equation:

$\frac{dy}{dx} = \frac{1}{8y+3}$

$\frac{dy}{dx} = \frac{1}{8(3)+3}$

$\frac{dy}{dx} = \frac{1}{27}$

## What is the reverse chain rule?

The reverse chain rule is used when integrating a function; it involves taking the differentiated function and taking it back to its original form.

Integrate $$\int{12(3x+3)^3 dx}$$

To do this, you can start by identifying your main function and breaking it down to revert it to its original integral. You can do this by working backwards:

$$12(3x + 3)^3$$

$$4(3x + 3)^3 \cdot 3$$

$$(3x + 3)^4$$

$$\int{12(3x + 3)^3 dx} = (3x + 3)^4 + c$$

Above is a breakdown of how to get to the answer. When differentiating x to a power, you can bring down the power in front of x and the power decreases by 1, for example x3 becomes 3x2. You also know that something has been multiplied together to get the 12 – in this instance, 4 since the power is 3. Taking it one step further back, you can take the 4 back up to a power. When using the reverse chain rule, it is also important that you add a constant to your answer, represented by c.

## Chain Rule - Key takeaways

• The chain rule is a rule used for differentiating composite functions, and these functions are also known as a function of a function.

• The formula that can be used when differentiating using the chain rule is:

$$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$$.

• The formula can also be written in function notation, if $$y = f'(g(x))$$ then $$\frac{dy}{dx} = f'(g(x))g'(x)$$

• The chain rule can also be used if the composite function involves trigonometric functions.

#### Flashcards in Chain Rule 2

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What is the chain rule?

The chain rule is a rule used in differentiating functions.

When do you use the chain rule?

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

What is the reverse chain rule?

The reverse chain rule is used when integrating a function, it involves taking a differentiated function back to its integral.

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