Learning Outcomes

  1. State the chain rule for the composition of two functions.
  2. Apply the chain rule together with other rules.
  3. Recognize the chain rule for a composition of three or more functions.


  • Chapter 3.6   The Chain Rule  

Textbook Assignment

  • p. 297:    215, 221, 222, 229-237 odd

WeBWork Assignment

  • Derivatives-Chain Rule
  • Derivatives-Higher Order

Exit problems of the session 

  1. Find the derivative of the following functions:

    (a).  $y=(3x^3-2x+1)^4$    (b).   $y=x^2\sin^4 x$   (c).  $y=\tan^2(3x)$
  2. Find the second order derivative $y”$ for the given function $y=\cos(x^2)$.


Key Concepts

  • The chain rule allows us to differentiate compositions of two or more functions. It states that for  $h(x)=f(g(x))$,

$h'(x)=f'(g(x)\cdot g'(x)$

         In Leibniz’s notation, this rule takes the form:


  • The chain rule can be combined with other rules we have learned. For example combining the chain rule with the power rule gives:

$\dfrac{d}{dx}(g(x))^n=n(g(x))^{n-1}\cdot g'(x)$ 

        or in Leibniz’s notation: 



Videos and Practice Problems of Selected Topics

  1. $\rhd$ Chain rule introduction  (5:06) Differentiate: $h(x) =\sin^2(x)$.
  2. * Practice: Chain rule. (6 problems with a guiding text)
  3. $\rhd$ Identifying composite functions (8:12) Find $f(g(x))$ when $f(x) =1+x$ and $g(x) = \cos(x)$. Next, identify $g(x) = \cos(\sin(x)+1)$ as the composition of functions. What about $f(x) =\sin(x)\cos(x)$? Is it a composition of functions?
  4. * Practice: Identify composite of functions. (4 problems)
  5. $\rhd$ Derivative of $\cos^3(x)$ using the chain rule. (9:16)
  6. $\rhd$ Derivative of $\sqrt{3x^2-x}$ using the chain rule. (5:30)
  7. * Practice: Chain rule intro. (4 problems)
  8. $\rhd$ Chain rule with table (2:26) For $F(x)=f(g(x))$, find $F'(4)$ given the values of $f(x)$, $g(x)$, $f'(x)$ and $g'(x)$ when $x=-2$ and $4$.
  9. * Practice: Chain rule with tables. (4 problems)