**Summary**

- Chain rule lets us differentiate a function of a function i.e
- Chain rule can be applied through two different equations:

AND

#### What is the chain rule?

One of the rules of differentiation is the chain rule, which basically lets us differentiate a **function of a function** in other words differentiation of composite functions.

Let’s suppose we have a function y which is formed by two different functions *f(x)* and *g(x)* i.e . *f(x)* is said to be the outer function and *g(x) i*s the inside function.

You will be able to understand this better when we go through an example.

#### Example #1

Let

We can see that in this case the outer function *f(x)* is and the inside function *g(x)* is .

When we put *g(x)* inside the expression of *f(x)* , like we get:

#### Example #2

As is actually

Therefore:

Hence:

Now that we have understood what it means by a ** ”function of a function”** or composite function, we can move on to the chain rule technique.

If we are able to differentiate function *g* with respect to *x* and we are able to differentiate function *f* with respect to *g(x)*, then the composite function is differentiable and we can use the chain rule technique to find its derivative.

The chain rule formula is defined as:

Where *g'(x)* is the derivative of the function *g(x)*,

and is the derivative of the function .

The following three steps explains how the formula is applied:

i) Differentiate the outside function *f(x)*

ii) Leave the inside function *g(x)* alone

iii) Multiply by the derivative of the inside function *g(x)*

There is also another way to apply the chain rule:

to find the derivative of we take *g(x)* and substitute it as *u* i.e:

Let *u = g(x)*

Therefore now* y = f(u).*

Hence, to find we can apply chain rule formula in a way that it becomes:

After we go through a few examples using both the formulas, you can choose which one you find easy to apply.

#### Example #3

Q. Find the derivative of

We know *f(x) = cos(x)*

Using equation 1 here:

Follow the steps to obtain:

i) *f'(x) = -sin(x)*

ii)

iii) *g'(x) = 2x*

Put the above values in the equation:

**Ans: **

#### Example #4

Q. Now solve the same question using **equation 2**

As

Let

Therefore:

* y = cos u*

Use equation 2:

To find we first find the following two:

and

Substitute the values in the equation:

substitute

Hence, this shows we reach the same answer using both the equations, so now you can decide which one is easier for you.