Essential Concepts

• The chain rule allows us to differentiate compositions of two or more functions. It states that for [latex]h(x)=f(g(x))[/latex],
  [latex]h^{\prime}(x)=f^{\prime}(g(x))g^{\prime}(x)[/latex]
  • In Leibniz’s notation this rule takes the form
    [latex]\frac{dy}{dx}=\frac{dy}{du} \cdot \frac{du}{dx}[/latex]
• 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 [latex]h(x)=(g(x))^n[/latex], then [latex]h^{\prime}(x)=n(g(x))^{n-1}g^{\prime}(x)[/latex]
• When applied to the composition of three functions, the chain rule can be expressed as follows: If [latex]h(x)=f(g(k(x)))[/latex], then [latex]h^{\prime}(x)=f^{\prime}(g(k(x)))g^{\prime}(k(x))k^{\prime}(x)[/latex]