Chain rule

Summary

• $h$ is a composite function of $f$ and $g$ :

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

• 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

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

• Alternative notation: If $y=f(u)$ and $u=g(x)$ then

$$\frac{dy}{dx}= \frac{dy}{du} \frac{du}{dx}$$

• The chain rule is also called the c omposition law .