Learning Outcomes
- Compute derivatives using a combination of algebraic derivative rules.
Section 2.6 Differentiation strategy (DF6)
Subsection 2.6.1 Activities
Activity 2.6.1.
Consider the functions defined below:
\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?
(b)
What do you notice that is different about these two functions?
(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?
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:
- How is this function built algebraically? What kind of function is this? What is the big picture?
- Where do you start?
- Is there an easier or more convenient way to write the function?
- Are there products or quotients involved?
- Is this function a composition of two (or more) elementary functions? If so, what are the outside and inside functions?
- What derivative rules will be needed along the way?
Activity 2.6.3.
Consider the function \(f(x)=x^3\sqrt{3-8x^2}\text{.}\)
(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?
- Chain rule
- Power rule
- Product rule
- Quotient rule
- Sum/difference rule
(b)
What other rules would be needed along the way? Select all that apply.
- Chain rule
- Power rule
- Product rule
- Quotient rule
- Sum/difference rule
(c)
Write an outline of the steps needed if you were asked to take the derivative of \(f(x)\text{.}\)
Activity 2.6.4.
Consider the function \(f(x)= \left(\frac{\ln x}{(3x-4)^3} \right)^5\text{.}\)
(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?
- Chain rule
- Power rule
- Product rule
- Quotient rule
- Sum/difference rule
(b)
What other rules would be needed along the way? Select all that apply.
- Chain rule
- Power rule
- Product rule
- Quotient rule
- Sum/difference rule
(c)
Write an outline of the steps needed if you were asked to take the derivative of \(f(x)\text{.}\)
Activity 2.6.5.
Consider the function \(f(x)= \sin(\cos(\tan(2x^3-1)))\text{.}\)
(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?
- Chain rule
- Power rule
- Product rule
- Quotient rule
- Sum/difference rule
(b)
What other rules would be needed along the way? Select all that apply.
- Chain rule
- Power rule
- Product rule
- Quotient rule
- Sum/difference rule
(c)
Write an outline of the steps needed if you were asked to take the derivative of \(f(x)\text{.}\)
Activity 2.6.6.
Consider the function \(f(x)= \frac{x^2 e^x}{2x^3-5x+\sqrt{x}}\text{.}\)
(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?
- Chain rule
- Power rule
- Product rule
- Quotient rule
- Sum/difference rule
(b)
What other rules would be needed along the way? Select all that apply.
- Chain rule
- Power rule
- Product rule
- Quotient rule
- Sum/difference rule
(c)
Write an outline of the steps needed if you were asked to take the derivative of \(f(x)\text{.}\)
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.
(a)
\(f(x) = e^{5x}(x^2+7^x)^3\)(b)
\(f(x) = \left( \frac{3x + 1}{2x^{6} - 1} \right)^{ 5 } \)(c)
\(f(x) = \sqrt{\cos\left(2 \, x^{2} + x\right)}\)(d)
\(f(x) = \tan(xe^x)\)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.
(a)
\begin{equation*}
f(y) = \sqrt{\cos\left(6 \, y^{4} - 6 \, y\right)}
\end{equation*}
(b)
\begin{equation*}
g(t) = \left( \frac{5 \, t^{3} + 2}{4 \, t^{4} - 3} \right)^{ 4 }
\end{equation*}
(c)
\begin{equation*}
h(x) = -{\left(5 \, x^{4} - 7 \, x^{3}\right)}^{5} x^{\frac{1}{4}}
\end{equation*}
