Q. 2.5.1 - Calculus

Question

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. 

(a) True or False: $\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{\pi }}\right)=0$

(b) True or False: $\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{z}}\right)={\mathrm{e}}^{\mathrm{z}}$

(c) True or False: $\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{x}}\right)=\mathrm{lnx}$

(d) True or False: $\frac{\mathrm{d}}{\mathrm{dx}}\left(\ln \left|\mathrm{x}\right|\right)=\frac{1}{\left|\mathrm{x}\right|}$

(e) True or False: If f is an exponential function, then f ' is a constant multiple of f . 

(f) True or False: If f ' is a constant multiple of f , then f is an exponential function. 

(g) True or False: Logarithmic differentiation is required in order to differentiate complicated products and quotients. 

(h) True or False: Logarithmic differentiation is required in order to differentiate expressions that have a variable in both the base and the exponent. 

Step-by-Step Solution

Answer

a. The  $\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{\pi }}\right)=0$ assertion is correct.

b. The $\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{z}}\right)={\mathrm{e}}^{\mathrm{z}}$ assertion is correct.

c. The $\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{x}}\right)=\mathrm{lnx}$ assertion is incorrect.

d. The $\frac{\mathrm{d}}{\mathrm{dx}}\left(\ln \left|\mathrm{x}\right|\right)=\frac{1}{\left|\mathrm{x}\right|}$  assertion is incorrect.

e. The derivatives of all exponential functions have the property of being constant multiples of the original function, given assertion is correct

f. The derivatives of all exponential functions have the property of being constant multiples of the original function. As a result, the given assertion is correct.

g. The derivatives of all exponential functions are constant multiples of the original. As a result, the assertion is correct.

h. The derivatives of any exponential function are always constant multiples of the original function. As a result, the given assertion is correct.

Part (a) Step 1: Given information

The given function is $\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{\pi }}\right)=0$

The following concept was used:

$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{constant}\right)=0$

Part (a) Step 2: Calculations

Consider that, 

$\begin{array}{l}\frac{\mathrm{d}}{\mathrm{dx}}\left(\text{costant}\right)=0\\ \frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{\pi }}\right)=0\end{array}$

The given assertion is correct.

Part (b) Step 1: Given information

The given function is $\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{z}}\right)={\mathrm{e}}^{\mathrm{z}}$

The following concept was used:

$\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{x}}\right)={\mathrm{e}}^{\mathrm{x}}$

Part (b) Step 2: Calculations

Consider that,

$\begin{array}{l}\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{x}}\right)={\mathrm{e}}^{\mathrm{x}}\\ \frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{z}}\right)={\mathrm{e}}^{\mathrm{z}}\end{array}$

As a result, the given assertion is correct.

Part (c) Step 1: Given information

The given function is $\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{x}}\right)=\mathrm{lnx}$

The following concept was used:

$\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{x}}\right)={\mathrm{nx}}^{\mathrm{n}-1}$

Part (c) Step 2: Calculations

Consider that, 

Furthermore,

$\begin{array}{l}\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{x}}\right)=-{\mathrm{x}}^{-1-1}\\ \frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\mathrm{x}}\right)=-{\mathrm{x}}^{-2}\end{array}$

As a result, the given assertion is incorrect.

Part (d) Step 1: Given information

The given function is $\frac{\mathrm{d}}{\mathrm{dx}}\left(\ln \left|\mathrm{x}\right|\right)=\frac{1}{\left|\mathrm{x}\right|}$

The following concept was used:

$\frac{\mathrm{d}}{\mathrm{dx}}\left(\ln \left|\mathrm{x}\right|\right)=\frac{1}{\mathrm{x}}$

Part (d) Step 2: Calculations

Consider that,

Furthermore,

$\frac{\mathrm{d}}{\mathrm{dx}}\left(\ln \left|\mathrm{x}\right|\right)=\frac{1}{\mathrm{x}}$

As a result, the given assertion is incorrect.

Part (e) Step 1: Given information

The given exponential function is ${\mathrm{e}}^{\mathrm{x}}$

The following concept was used:

$\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{x}}\right)={\mathrm{e}}^{\mathrm{x}}$

Part (e) Step 2: Calculations

The derivatives of all exponential functions have the property of being constant multiples of the original function.

As a result, the given assertion is correct.

Part (f) Step 1: Given information

The given function is $\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{constant}$

The following concept was used:

$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{constant}\right)=0$

Part (f) Step 2: Calculations

The derivatives of all exponential functions are constant multiples of the original function.

As a result, the stated assertion is correct.

Part (g) Step 1: Given information

In order to differentiate complicated products and quotients, logarithmic differentiation is required.

The following concept was used:

$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{logx}\right)=\frac{1}{\mathrm{x}}$

Part (g) Step 2: Calculations

The derivatives of all exponential functions are constant multiples of the original function.

As a result, the stated assertion is correct.

Part (h) Step 1: Given information

To differentiate statements with a variable in both the base and the exponent, logarithmic differentiation is necessary.

The following concept was used:

$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{logx}\right)=\frac{1}{\mathrm{x}}$

Part (h) Step 2: Calculations

The derivatives of all exponential functions have the property of being constant multiples of the original function.

As a result, the given assertion is correct.