Learning Outcomes
- State the chain rule for the composition of two functions.
- Apply the chain rule together with other rules.
- Recognize the chain rule for a composition of three or more functions.
Textbook
- 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
-
Find the derivative of the following functions:
(a). $y=(3x^3-2x+1)^4$ (b). $y=x^2\sin^4 x$ (c). $y=\tan^2(3x)$ - 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:
$\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}$
- 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:
$\dfrac{d}{dx}u^n=nu^{n-1}\cdot\dfrac{du}{dx}$
Videos and Practice Problems of Selected Topics
- $\rhd$ Chain rule introduction (5:06) Differentiate: $h(x) =\sin^2(x)$.
- * Practice: Chain rule. (6 problems with a guiding text)
- $\rhd$ 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?
- * Practice: Identify composite of functions. (4 problems)
- $\rhd$ Derivative of $\cos^3(x)$ using the chain rule. (9:16)
- $\rhd$ Derivative of $\sqrt{3x^2-x}$ using the chain rule. (5:30)
- * Practice: Chain rule intro. (4 problems)
- $\rhd$ 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 $4$.
- * Practice: Chain rule with tables. (4 problems)
