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Calculus for Team-Based Inquiry Learning 2025 Edition

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Section 2.6 Differentiation Strategy (DF6)

Subsection 2.6.1 Activities

Activity 2.6.1.

Consider the functions defined below:
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\begin{equation*} f(x)=\sin((x^2+3x)\cos(2x)) \end{equation*}
\begin{equation*} g(x)=\sin(x^2+3x)\cos(2x) \end{equation*}
(a)
What do you notice that is similar about these two functions?
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(b)
What do you notice that is different about these two functions?
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(c)
Imagine that you are sorting functions into different categories based on how you would differentiate them. In what category (or categories) might these functions fall?
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Remark 2.6.2.

To take a derivative, we need to examine how the function is built and then proceed accordingly. Below are some questions you might ask yourself as you take the derivative of a function, especially one where multiple rules might need to be used:
  1. How is this function built algebraically? What kind of function is this? What is the big picture?
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  2. Where do you start?
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  3. Is there an easier or more convenient way to write the function?
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  4. Are there products or quotients involved?
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  5. Is this function a composition of two (or more) elementary functions? If so, what are the outside and inside functions?
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  6. What derivative rules will be needed along the way?
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Activity 2.6.3.

Consider the function \(f(x)=x^3\sqrt{3-8x^2}\text{.}\)
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(a)
You will need multiple derivative rules to find \(f'(x)\text{.}\) Which rule would need to be applied first? In other words, what is the big picture here?
  1. Chain rule
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  2. Power rule
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  3. Product rule
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  4. Quotient rule
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  5. Sum/difference rule
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(b)
What other rules would be needed along the way? Select all that apply.
  1. Chain rule
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  2. Power rule
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  3. Product rule
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  4. Quotient rule
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  5. Sum/difference rule
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(c)
Write an outline of the steps needed if you were asked to take the derivative of \(f(x)\text{.}\)
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Activity 2.6.4.

Consider the function \(f(x)= \left(\dfrac{\ln x}{(3x-4)^3} \right)^5\text{.}\)
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(a)
You will need multiple derivative rules to find \(f'(x)\text{.}\) Which rule would need to be applied first? In other words, what is the big picture here?
  1. Chain rule
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  2. Power rule
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  3. Product rule
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  4. Quotient rule
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  5. Sum/difference rule
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(b)
What other rules would be needed along the way? Select all that apply.
  1. Chain rule
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  2. Power rule
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  3. Product rule
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  4. Quotient rule
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  5. Sum/difference rule
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(c)
Write an outline of the steps needed if you were asked to take the derivative of \(f(x)\text{.}\)
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Activity 2.6.5.

Consider the function \(f(x)= \sin(\cos(\tan(2x^3-1)))\text{.}\)
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(a)
You will need multiple derivative rules to find \(f'(x)\text{.}\) Which rule would need to be applied first? In other words, what is the big picture here?
  1. Chain rule
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  2. Power rule
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  3. Product rule
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  4. Quotient rule
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  5. Sum/difference rule
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(b)
What other rules would be needed along the way? Select all that apply.
  1. Chain rule
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  2. Power rule
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  3. Product rule
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  4. Quotient rule
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  5. Sum/difference rule
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(c)
Write an outline of the steps needed if you were asked to take the derivative of \(f(x)\text{.}\)
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Activity 2.6.6.

Consider the function \(f(x)= \dfrac{x^2 e^x}{2x^3-5x+\sqrt{x}}\text{.}\)
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(a)
You will need multiple derivative rules to find \(f'(x)\text{.}\) Which rule would need to be applied first? In other words, what is the big picture here?
  1. Chain rule
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  2. Power rule
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  3. Product rule
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  4. Quotient rule
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  5. Sum/difference rule
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(b)
What other rules would be needed along the way? Select all that apply.
  1. Chain rule
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  2. Power rule
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  3. Product rule
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  4. Quotient rule
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  5. Sum/difference rule
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(c)
Write an outline of the steps needed if you were asked to take the derivative of \(f(x)\text{.}\)
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Activity 2.6.7.

Find the derivative of the following functions. For each, include an explanation of the steps involved that references the algebraic structure of the function.
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(a)
\(f(x) = e^{5x}(x^2+7^x)^3\)
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(b)
\(f(x) = \left( \dfrac{3x + 1}{2x^{6} - 1} \right)^{ 5 } \)
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(c)
\(f(x) = \sqrt{\cos\left(2 \, x^{2} + x\right)}\)
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Activity 2.6.8.

Demonstrate and explain how to find the derivative of the following functions. Be sure to explicitly denote which derivative rules (constant multiple, sum/difference, etc.) you are using in your work.
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(a)
\begin{equation*} f(y) = \sqrt{\cos\left(6 \, y^{4} - 6 \, y\right)} \end{equation*}
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(b)
\begin{equation*} g(t) = \left( \frac{5 \, t^{3} + 2}{4 \, t^{4} - 3} \right)^{ 4 } \end{equation*}
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(c)
\begin{equation*} h(x) = -{\left(5 \, x^{4} - 7 \, x^{3}\right)}^{5} x^{\frac{1}{4}} \end{equation*}
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Subsection 2.6.2 Videos

Figure 54. Video for DF6
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Subsection 2.6.3 Exercises

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