Chain rule

Summary

\[y=h\left( x \right)=f\left( g\left( x \right) \right)=(f∘g)(x)\]

we denote \(u=g(x)\) such that \(h(x)=f(u)\) where \(g\left( x \right)\) is the inner function and \(y=f\left( u \right)\) is the outer function.
Example: \(y=h(x)=exp \left( x^2 \right)\) then \(h=f∘g\) where \(u=g(x)=x^2\) and \(y=f(u)=e^u\) .
If \(g\) is differentiable at \(x\) and \(f\) is differentiable at \(u=g(x)\) then \(h=f∘g\) is differentiable at \(x\) . We have

\[h'\left( x \right)=f'(u)g'(x)=f'\left( g\left( x \right) \right)g'(x)\]

\(f'\left( u \right)=f'\left( g\left( x \right) \right)\) is called the outer derivative and \(g'\left( x \right)\) is called the inner derivative.
Example: \(y=h(x)=exp \left( x^2 \right)\) . \(f'\left( u \right)=e^u=exp \left( x^2 \right)\) and \(g'\left( x \right)=2x\) and

\[h'\left( x \right)=exp \left( x^2 \right)⋅2x\]

\[ \frac{dy}{dx}= \frac{dy}{du} \frac{du}{dx}\]