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: Determine the derivative of \( 5^x \).
Rewrite \( 5^x \) in exponential form:\[5^x = e^{x \log 5}.\]Differentiate \( 5^x \) 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: Determine the derivative of \( e^x \).
The derivative of \( e^x \) with respect to \( x \) is:\[\frac{d}{dx}(e^x) = e^x.\]Step 3: Calculate the derivative of \( 5^x \) with respect to \( e^x \).
Apply 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 expression.
Substitute \( 5^x = \left( \frac{5}{e} \right)^x \cdot e^x \) into the derivative:\[\frac{5^x \cdot \log 5}{e^x} = \left( \frac{5}{e} \right)^x \cdot \log 5.\]The final result is:\[\left( \frac{5}{e} \right)^x \cdot \log 5.\]