Q. 1C - Calculus

Question

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

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

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

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

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

(f) True or False: If $f\text{'}$ 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

Part (a). The given statement is true.

Part (b). The given statement is true.

Part (c). The given statement is false.

Part (d). The given statement is false.

Part (e). The given statement is true.

Part (f). The given statement is true.

Part (g). The given statement is true.

Part (h). The given statement is true.

Part (a). Step 1. Explanation

Concept used: $\frac{d}{dx}\left(cons\tan t\right)=0$

Because, $$\begin{gathered} \frac{d}{dx}\left(cons\tan t\right)=0 \\ \frac{d}{dx}\left(e^{\mathrm{\pi }}\right)=0 \end{gathered}$$

Hence, the given statement is true.

Part (b) Step 1. Explanation

Concept used: $\frac{d}{dx}\left(e^x\right)=e^x$

Since, $$\begin{gathered} \frac{d}{dx}\left(e^x\right)=e^x \\ \frac{d}{dz}\left(e^z\right)=e^z \end{gathered}$$

Hence, the given statement is true.

Part (c) Step 1. Explanation

Concept used: $\frac{d}{dx}\left(\frac{1}{x}\right)=n{x}^{n-1}$

False

Since, $$\begin{gathered} \frac{d}{dx}\left(\frac{1}{x}\right)=-x^{-1-1} \\ \frac{d}{dx}\left(\frac{1}{x}\right)=-x^{-2} \end{gathered}$$

Hence, the given statement is false.

Part (d) Step 1. Explanation

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

False

Since, $$\begin{gathered} \frac{d}{dx}\left(\ln \left|x\right|\right)\frac{1}{x} \\ \frac{d}{dx}\left(\ln \left|x\right|\right)\frac{1}{x} \end{gathered}$$

Hence, the given statement is false.

Part (e) Step 1. Explanation

Concept used: $\frac{d}{dx}\left(e^x\right)=e^x$

True, 

All exponential function have the property that their derivatives are constant multiples of the original function.

Thus the given statement is true.

Part (f) Step 1. Explanation

Concept used: $\frac{d}{dx}\left(cons\tan t\right)=0$

True, 

All exponential function have the property that their derivatives are constant multiples of the original function.

Thus the given statement is true.

Part (g) Step 1. Explanation

Concept used: $\frac{d}{dx}\left(\log x\right)=\frac{1}{x}$

True, 

All exponential function have the property that their derivatives are constant multiples of the original function.

Thus the given statement is true.

Part (h) Step 1. Explanation

Concept used: $\frac{d}{dx}\left(\log x\right)=\frac{1}{x}$

True, 

All exponential function have the property that their derivatives are constant multiples of the original function.

Thus the given statement is true.