    # 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(√  )
So y = log u  where u = √
The above functions cannot be differentiated directly, so we introduce a new rule here.

Chain Rule

Consider a composite function y = F(u) where u   is the function u = f(x).

If F(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.

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