Question

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Functions f and g, which illustrate that, in general, derivatives and products do not commute.

(b) Functions f and g, which illustrate that, in general, derivatives and quotients do not commute.

(c) Three functions whose derivatives we cannot calculate by using the differentiation rules we have developed so far.

Short answer

(a) Derivatives and products do not commute

(b)derivatives and quotients do not commute.

(c)Three functions whose derivatives we cannot calculate by using the differentiation rules

$$\begin{gathered} f\left(x\right)=\sin \left(x^2\right) \\ f\left(x\right)=\sqrt{x^5} \\ f\left(x\right)=e^{\sin \left(x\right)} \end{gathered}$$

Step-by-step solution

Given Information 

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

Part(a) : Step 1: Commute 

Functions f and g, which illustrate that, in general, derivatives and products do not commute.

$$\begin{gathered} Forexample \\ f\left(x\right)=x^3+1 \\ g\left(x\right)=x^4 \\ f\text{'}\left(x\right)=3x^2 \\ g\text{'}\left(x\right)=4x^3 \\ derivative \\ f\text{'}\left(x\right)*g\text{'}\left(x\right)=\left(3x^2\right)\left(4x^3\right)=12x^5 \\ f\left(x\right)g\left(x\right)=(x^3+1)\left(x^4\right)=x^7+x^4 \\ f\text{'}\left(x\right)g\text{'}\left(x\right)\ne f\left(x\right)g\left(x\right) \end{gathered}$$

derivatives and products do not commute

Part(b) : Step (1) commute

Functions f and g, which illustrate that, in general, derivatives and quotients do not commute.

$$\begin{gathered} f\left(x\right)=x^2+1 \\ g\left(x\right)=x^4 \\ f\text{'}\left(x\right)=2x \\ g\text{'}\left(x\right)=4x^3 \\ \frac{f\text{'}\left(x\right)}{g\text{'}\left(x\right)}=\frac{2x}{4x^3} \\ \frac{f\left(x\right)}{g\left(x\right)}=\frac{x^2+1}{x^4} \\ \frac{f\text{'}\left(x\right)}{g\text{'}\left(x\right)}\ne \frac{f\left(x\right)}{g\left(x\right)} \end{gathered}$$

derivatives and quotients do not commute.

Part(c) Step 1: functions

Three functions whose derivatives we cannot calculate by using the differentiation rules we have developed so far.

$$\begin{gathered} f\left(x\right)=\sin \left(x^2\right) \\ f\left(x\right)=\sqrt{e^x} \\ f\left(x\right)=e^{\sin \left(x\right)} \end{gathered}$$