3.2: Move Those Chains
This activity is intended to supplement Calculus, Chapter 2, Lesson 5.
ID: 11364
Time Required: 15 minutes
Activity Overview
In this activity, students will explore the Chain Rule. Students are first asked to make a conjecture about the derivative of \begin{align*}f(x) = (2x + 1)^2\end{align*}
Topic: Chain Rule
 Derivative of a composite function
Teacher Preparation and Notes

Students will type \begin{align*}d(f(x),x)\end{align*}
d(f(x),x) in the entry line of the TI89. When they press ENTER, the TI89 will return the expression in ‘pretty print’, and it will appear as \begin{align*}\frac{d}{dx}(f(x))\end{align*}ddx(f(x)) . 
Note: Some functions will have an independent variable other than \begin{align*}x\end{align*}
x to familiarize students with using other variables.  The true statements show the derivatives in unsimplified form so that students will more easily identify the patterns.
Associated Materials
 Student Worksheet: Move those Chains http://www.ck12.org/flexr/chapter/9727, scroll down to the second activity.
Problem 1
Students are asked to make a conjecture about what the derivative of \begin{align*}f(x) = (2x + 1)^2\end{align*}
Students are asked to expand \begin{align*}(2x + 1)^2\end{align*}
Solutions
1. Sample answer: \begin{align*}2(2x + 1)\end{align*}
2. Sample answer: No, my answer was not correct. I can try expanding the function before taking the derivative.
3.\begin{align*}\frac{d}{dx}((2x + 1)^2) & = \frac{d}{dx}((2x + 1)(2x + 1)) \\
& = \frac{d}{dx}(4x^2 + 4x + 1) \\
& = \frac{d}{dx}(4x^2) + \frac{d}{dx}(4x) + \frac{d}{dx}(1) \\
& = 8x + 4\end{align*}
Problem 2
Students are asked to examine “true” statements of the derivatives of composite functions while looking for patterns. They are asked to discuss the patterns they observed with fellow students.
Students are asked to use the pattern observed to make “true” derivative of composite function statements. If the handheld does not return the word ‘true,’ students can try again by pressing \begin{align*}2^{nd}\end{align*}
The Chain Rule is presented to students, and they are asked to write three additional true statements.
Solutions
4. Sample answer: The Power Rule is applied to the “outer” function, then is multiplied by the derivative of the “inner” function.
5. \begin{align*}\frac{d}{dx}((3x + 2)^2) = 2 \cdot (3x + 2)^1 \cdot 3\end{align*}
6. \begin{align*}\frac{d}{dx}((7x + 2)^3) = 3 \cdot (7x + 2)^2 \cdot 7\end{align*}
7. \begin{align*}\frac{d}{dx}((5x^2 + 2x + 3)^4) = 4 \cdot (5x^2 + 2x + 3)^3 \cdot (10x + 2)\end{align*}
8. Answers may vary.
Problem 3 – Homework Problems
Students are given five additional exercises that can be used as homework problems or as extra practice during class.
Solutions
1. \begin{align*}\frac{d}{dx}((4x^3 + 1)^2) = 2 \cdot (4x^3 + 1)^1 \cdot (12x^2)\end{align*}
2. \begin{align*}\frac{d}{dx}((5x + 10)^7) = 7 \cdot (5x + 10)^6 \cdot (5)\end{align*}
3. \begin{align*}\frac{d}{dx}((2t^5  4t^3 + 2t  1)^2) = 2 \cdot (2t^5  4t^3 + 2t  1)^1 \cdot (10t^4  12t^2 + 2)\end{align*}
4. \begin{align*}\frac{d}{dx}((x^2 + 5)^{2}) = 2 \cdot (x^2 + 5)^{3} \cdot 2x\end{align*}
5. \begin{align*}\frac{d}{dz}((z^3  3z^2 + 4)^{3}) = 3 \cdot (z^3  3z^2 + 4)^{4} (3z^2  6z)\end{align*}