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# 3.2: Move those Chains

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Calculus, Chapter 2, Lesson 5.

## Problem 1 – Derivative Using the Power Rule

Recall the Power Rule $\frac{d}{dx}(x^n)=n \cdot x^{n-1}$.

1. Based on the Power Rule, what do you think the derivative of $f(x) = (2x + 1)^2$ is?

Graph the derivative of the function and your conjecture about the derivative. Go to the $Y=$ Editor. In $y1$, type $(2x+1)^{\land}2$. In $y2$, type nDeriv$(y1(x),x)$. To access the nDeriv command, go to the Math menu ($2^{nd}$ [MATH]) and select B:Calculus > A:nDeriv(. In $y3$, type your conjecture for the derivative of $f(x) = (2x + 1)^2$. Highlight $y1$ and press $F4$ to unselect this function, and press $[\blacklozenge]$ $F3$ to graph $y2$ and $y3$. Note: The graph may take a minute to appear. If your conjecture is correct, the graphs of $y2$ and $y3$ will coincide. If your conjecture is incorrect, the graphs of $y2$ and $y3$ will not coincide.

2. Was your conjecture correct? If not, how can you change your conjecture to make it correct?

3. Expand the binomial $(2x + 1)^2$. Take the derivative of each term. How does this compare with your answer to Question 1?

## Problem 2 – The Chain Rule

The following are ‘true’ statements that can be verified on the TI-89.

$d((5x + 7)^\land 3,x)=3 \cdot(5x + 7)^\land 2 \cdot 5x$ true

$d((x^\land 3 + 7)^\land 5, x)=5 \cdot(x^\land 3 + 7)^\land 4 \cdot 3x^\land 2$ true

$d((x^\land 2 + 6)^\land 4, x)=4 \cdot(x^\land 2 + 6)^\land 3 \cdot 2x$ true

4. What patterns do you see? Using any information that you can infer from these statements, create a rule for finding the derivative of these functions. Discuss the patterns you see and the rule you created with a partner.

5. Using your rule from Question 4, what is $\frac{d}{dx} \left((3x+2)^2 \right)$?

Verify your answer by typing your statement on the entry line of your TI-89. If you are correct, the TI-89 will return the word, ‘true’. If you are incorrect, the TI-89 will return a false statement. If you are incorrect, try again by editing your statement. You can copy your last command by selecting $2^{nd}$ ENTER.

6. What is $\frac{d}{dx}((7x+2)^3)$? Verify your answer.

7. What is $\frac{d}{dx}((4x^2 + 2x + 3)^4)$? Verify your answer.

The derivative rule you have just observed is called the Chain Rule. It is used to take the derivative of composite functions. The Chain Rule is $\frac{d}{dx}(f(g(x)))=f'(g(x)) \cdot g'(x)$. First, take the derivative of the “outside function” at $g(x)$. Then, multiply this by the derivative of the “inside function.”

8. Use the Chain Rule to create three additional true statements. Verify your answers.

## Problem 3 – Homework Problems

Evaluate the following derivatives using the Chain Rule. Verify your answers.

1. $\frac{d}{dx}((4x^3 + 1)^2)=$
2. $\frac{d}{dx}((-5x + 10)^7)=$
3. $\frac{d}{dt}((2t^5 - 4t^3 + 2t -1)^2)=$
4. $\frac{d}{dx}((x^2 + 5)^{-2})=$
5. $\frac{d}{dz}((z^3 - 3z^2 + 4)^{-3})=$

## Date Created:

Feb 23, 2012

Nov 04, 2014
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