Question

The derivative of $5^x$ w.r.t. $e^x$ is: {5pt}

• A. $\left( \frac{5}{e} \right)^x \frac{1}{\log 5}$
• B. $\left( \frac{e}{5} \right)^x \frac{1}{\log 5}$
• C. $\left( \frac{5}{e} \right)^x \log 5$
• D. $\left( \frac{e}{5} \right)^x \log 5$

Hint:

When differentiating exponential functions: - Use $a^x = e^{x \log a}$ for conversion. - The derivative of $a^x$ is $a^x \cdot \log a$. - When differentiating with respect to another function, apply the chain rule.

Answer 1

The Correct Option is C

Step 1: Express the derivative of $5^x$.
The function $5^x$ can be written in exponential form as: $$5^x = e^{x \log 5}.$$ Differentiating $5^x$ with respect to $x$: $$\frac{d}{dx}(5^x) = \frac{d}{dx}(e^{x \log 5}) = e^{x \log 5} \cdot \log 5 = 5^x \cdot \log 5.$$ Step 2: Express the derivative of $e^x$.
The derivative of $e^x$ with respect to $x$ is: $$\frac{d}{dx}(e^x) = e^x.$$ Step 3: Find the derivative of $5^x$ with respect to $e^x$.
Using the chain rule: $$\frac{d}{d(e^x)}(5^x) = \frac{\frac{d}{dx}(5^x)}{\frac{d}{dx}(e^x)} = \frac{5^x \cdot \log 5}{e^x}.$$ Step 4: Simplify the result.
Since $5^x = \left( \frac{5}{e} \right)^x \cdot e^x$, substituting this into the derivative gives: $$\frac{5^x \cdot \log 5}{e^x} = \left( \frac{5}{e} \right)^x \cdot \log 5.$$ Thus, the final answer is: $$\left( \frac{5}{e} \right)^x \cdot \log 5.$$