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Chain Rule

Chain Rule Integration Assignment / Homework Help

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:

Chain Rule Integration Assignment / Homework Help

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:

Chain Rule Integration Assignment / Homework Help


Chain Rule Integration Assignment / Homework Help

Chain Rule Integration Assignment / Homework Help


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
Chain Rule Integration Assignment / Homework Help


Here the outermost function is the exponential function

Chain Rule Integration Assignment / Homework Help

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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