Q2 - Calculus

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.

Step-by-Step Solution

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 1: 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. 

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

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. 

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