Chain Rule

Before moving to our last derivative result, the chain rule, you might consider reviewing the Appendix on function composition, F . The chain rule is considered the most challenging of the three results, the produce rule, quotient rule, and chain rule, partly due to function composition itself being confusing. We are going to use $$\sin (x^2)$$ sin ( x 2 ) as a main example, which is a composition of $$\sin (x)$$ sin ( x ) and $$x^2$$ x 2 . In other words, if $$f(x)= \sin (x)$$ f ( x ) = sin ( x ) and $$g(x)=x^2$$ g ( x ) = x 2 then $$h(x)=f(g(x)) = \sin (x^2)$$ h ( x ) = f ( g ( x ) ) = sin ( x 2 ) . In this example, $$g(x)=x^2$$ g ( x ) = x 2 is the inside function while $$f(x)=\sin (x)$$ f ( x ) = sin ( x ) is the outside function, because g(x) is inside f(x) in $$h(x)=f(g(x))$$ h ( x ) = f ( g ( x ) ) . Now, if we are given $$h(x)= = \sin (x^2)$$ h ( x ) = = sin ( x 2 ) how do we know this is a composition and how do we know which is the inside function. What may help it to consider what we would do if we were to evaluate h(x) at some value of x, say $$x=5$$ x = 5 . We would first do $$5^2$$ 5 2 to get 25 after that we would then evaluate $$\sin (25)$$ sin ( 25 ) . Here $$x^2$$ x 2 is the inside function because we did that first, squared 5, and then took that output and used it to find $$\sin (25)$$ sin ( 25 ) , making $$\sin (x)$$ sin ( x ) the outside function. We will consider both how function composition changes functions along with the impact on the derivative.