# Chain Rule

Introduction

We use the chain rule to find the derivative of a composite function.

Consider a function

*y*= F (*u*) where u = f(*x*) is a function of*x*. then we say that*y*is a composite function of*x*with intermediate argument*u*.
For example:

- y = sin(
*x*2 )

Here we can write*y*= sin*u*where*u*=*x*2 *y*= log(√*x*)

So*y*= log*u*where*u*= √*x*

**Chain Rule**

Consider a composite function

*y*= F(*u*) where*u*is the function*u*= f(*x*).
If F(

Where in place of

We can also read this rule as:

*u*) has a derivative F'(*u*) and f(x) has a derivative f '(*x*) then*y*is differentiable with respect to*x*and its derivative is given by*y*’(*x*) = F’(*u*). f ’(*x*)Where in place of

*u*we must substitute*u*= f(*x*).We can also read this rule as:

This rule can also be extended to a composition of more than two functions.

So if

*y*= F (

*u*),

*u*= f (

*t*) and

*t*= g (

*x*) we get the derivative of y as follows:

Examples:

**Note:**

We can remember this rule in following two simple steps:

- Take the derivative of the outermost function
- Multiply by the derivative of the inner function

Solution:

Here the outermost function is the exponential function

As we can see above, chain rule can be applied repeatedly starting from the outermost function to get the derivative of any complex composite function.

As we can see above, chain rule can be applied repeatedly starting from the outermost function to get the derivative of any complex composite function.

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