Summary of the Chain Rule

The chain rule allows us to differentiate compositions of two or more functions. It states that for h(x)=f(g(x)) $h (x) = f (g (x))$ ,
$h^{\prime}(x)=f^{\prime}(g(x))g^{\prime}(x)$
- In Leibniz’s notation this rule takes the form
  
  $\frac{dy}{dx}=\frac{dy}{du} \cdot \frac{du}{dx}$
We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them.
The chain rule combines with the power rule to form a new rule:
If $h(x)=(g(x))^n$ , then $h^{\prime}(x)=n(g(x))^{n-1}g^{\prime}(x)$
When applied to the composition of three functions, the chain rule can be expressed as follows: If $h(x)=f(g(k(x)))$ , then $h^{\prime}(x)=f^{\prime}(g(k(x)))g^{\prime}(k(x))k^{\prime}(x)$

Key Equations

The chain rule
$\frac{d}{dx}(f(g(x)))=f^{\prime}(g(x))g^{\prime}(x)$
The power rule for functions
$\frac{d}{dx}((g(x)^n)=n(g(x))^{n-1}g^{\prime}(x)$

chain rule: the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function