91Ó°ÊÓ

101-107. Choose the correct answer: $$ \frac{d}{d x} e^{x}=\quad \text { a. } x e^{x-1} \quad \text { b. } e^{x} \quad \text { c. } 0 $$

101-107. Choose the correct answer: $$ \begin{aligned} &\frac{d}{d x} e^{f(x)}=\quad \text { a. } e^{f(x)} f^{\prime}(x)\\\ &\text { b. } e^{f^{\prime}(x)} f(x) \text { c. } e^{f^{\prime}(x)} f^{\prime}(x) \end{aligned} $$

101-107. Choose the correct answer: $$ \frac{d}{d x} e^{5}=\quad \text { a. } 5 e^{4} \quad \text { b. } e^{5} \quad \text { c. } 0 $$

101-107. Choose the correct answer: $$ \frac{d}{d x} x^{e}=\quad \text { a. } e x^{e-1} \quad \text { b. } x^{e} \quad \text { c. } 0 $$

101-107. Choose the correct answer: $$ \frac{d}{d x} \ln x=\quad \text { a. } \frac{x}{1} \quad \text { b. } \frac{1}{x} $$

101-107. Choose the correct answer: $$ \begin{aligned} &\frac{d}{d x} \ln f(x)=\text { a. } \frac{f(x)}{f^{\prime}(x)} \quad \text { b. } \frac{f^{\prime}(x)}{f(x)}\\\ &\text { c. } \frac{f^{\prime}(x)}{f^{\prime}(x)} \end{aligned} $$

101-107. Choose the correct answer: $$ \frac{d}{d x} \ln 5=\quad \text { a. } \frac{5}{1} \quad \text { b. } \frac{1}{5} \quad \text { c. } 0 $$

True or False: If a function involves natural logarithms, then its derivative will involve natural logarithms. Explain.

Explain why it is obvious, without any calculation, that \(\frac{d}{d x} e^{\ln x}=1\).

Explain why it is obvious, without any calculation, that \(\frac{d}{d x} \ln e^{x}=1\)