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In this section, we will cover the aspects of the Chain Rule which are derivatives of composite functions
Some examples of composite function are:
) = (x+1)^2 "f(g(x)) = (x+1)^2")
where =x^2 "f(x)=x^2") and =x+1 "g(x)=x+1")
or
)=sin(cos(x)) "f(g(x))=sin(cos(x))")
where =sin(x) "f(x)=sin(x)") and =cos(x) "g(x)=cos(x)")
Chain Rule
Given a function
 = f(g(x)) "F(x) = f(g(x))")
the derivative of this function is
 = f\prime(g(x)) g\prime(x) "F\prime(x) = f\prime(g(x)) g\prime(x)")
Sometimes this is described as "the derivative of the outside multiplied by the derivative of the inside", although that is a somewhat general statement and doesn't precisely explain how to use the Chain rule. Some examples will be beneficial in our understanding of this principle.
Examples
Find the derivative of the following functions
=(x+1)^2 "1. \,\,F(x)=(x+1)^2")
We can see that this is a composite function where
and
the derivatives of these functions are
=2x "f\prime (x)=2x") and =1 "g\prime (x)=1")
since
then
 =f\prime(g(x)) g\prime(x) =f\prime(x+1)(1) = 2(x+1) "F\prime(x) =f\prime(g(x)) g\prime(x) =f\prime(x+1)(1) = 2(x+1)")
=sin(cos(x)) "2. \,\, F(x)=sin(cos(x))")
We can see that this is a composite function where
and
the derivatives of these functions are
=cos(x) "f\prime(x)=cos(x)") and  = -sin(x) "g\prime(x) = -sin(x)")
since
then
 =f\prime(g(x)) g\prime(x) =f\prime(cos(x))(-sin(x)) = cos(cos(x))(-sin(x)) "F\prime(x) =f\prime(g(x)) g\prime(x) =f\prime(cos(x))(-sin(x)) = cos(cos(x))(-sin(x))")
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