Question

If \( y = e^{\log x} \), then \( \frac{dy}{dx} \) is:

• A. \( \log x - x \)
• B. \( x e^{\log x} \)
• C. 1
• D. \( e^{\log x} \log x \)

Hint:

Always simplify the expression using logarithmic properties before starting the differentiation process. It converts a complex-looking chain rule problem into a basic one.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Logarithmic and exponential functions (with the same base) are inverse operations. They cancel each other out.
Step 2: Detailed Explanation:
Given the function \( y = e^{\log_e x} \).
By the property of logarithms \( a^{\log_a f(x)} = f(x) \), the expression simplifies to:
\[ y = x \]
Now, we differentiate with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x) \] \[ \frac{dy}{dx} = 1 \]
Step 3: Final Answer:
The derivative is 1.