Calculus U5L10 - Review of Composite Functions

Identify the  "composite" function.

• $y=x$

• $y=x^2$

•  $y=\left(x+3\right)^5$ 

•  $y=x+4$ 

•  $y=2x$  

Identify the "composite" function.

• $y=\sqrt{2x}$

• $y=\sqrt{x}$

• $y=x^{\frac{1}{3}}$

• $y=x^4$

Identify the "composite" functions.  Select ALL that apply.

•  $y=\tan x$ 

•  $y=\sin x$ 

•  $y=\ \cos3x$ 

•  $y=\sin\left(x+5\right)$  

Identify the "composite" functions.  Select ALL that apply.

•  $y=\left(3x+4\right)^8$ 

•  $y=\left(1+x^4\right)^2$ 

•  $y=\ x+1$ 

•  $y=\sqrt{x^2+3}$  

Using the first four questions you just answered, which of the following is the best definition of a composite function.

• A COMPOSITE FUNCTION is a function that uses only one operation on the independent variable to obtain the output.

• A COMPOSITE FUNCTION is a function that uses more than one operation on the independent variable to obtain the output.

• A COMPOSITE FUNCTION is a function that uses exponents on the independent variable to obtain the output.

Assess your understanding of composite functions with respect to:

• I can recognize composite functions.
• I can define what a composite function is.

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In order to integrate composite functions, we need to review taking the derivative of composite functions. To take the derivative of a composite function, we use the CHAIN RULE. In your own words, explain how to use the chain rule to take the derivative of a composite function.

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The CHAIN RULE is used to take the derivative of composite functions. To use it, take the derivative of each "layer" while keeping the "inside" function the same. How many "layers" does this function $f\left(x\right)=\left(3x^4-5\right)^7$ have? 

• 1

• 2

• 3

For the composite function $f\left(x\right)=\left(3x^4-5\right)^7$ , select the "inside" function AND its corresponding derivative. 

•  $3x^4-5$  

•  $\left(\ \ \right)^7$  

•  $12x^3$  

•  $7\left(\ \ \ \right)^6$  

For the composite function $f\left(x\right)=\left(3x^4-5\right)^7$ , select the "outside" function AND its corresponding derivative. 

•  $3x^4-5$  

•  $\left(\ \ \right)^7$  

•  $12x^3$  

•  $7\left(\ \ \ \right)^6$