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)