Question

The derivative of \(x\log_e(x)\) is

• A. \(1\)
• B. \(0\)
• C. \(e^x\)
• D. \(1+\log_e(x)\)

Hint:

Whenever a function is a product of two terms, always check whether the product rule of differentiation is required.

Answer 1

The Correct Option is D

Step 1: Spot the product.
We differentiate $y = x\,\log_e x$, a product of $x$ and $\log_e x$, so the product rule fits.

Step 2: Write the rule.
For $y = uv$, \[ \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx} \] with $u=x$ and $v=\log_e x$.

Step 3: Differentiate each piece.
Here $\dfrac{du}{dx} = 1$ and $\dfrac{dv}{dx} = \dfrac{1}{x}$.

Step 4: Plug in and tidy.
\[ \frac{dy}{dx} = x\cdot\frac{1}{x} + \log_e x \cdot 1 = 1 + \log_e x \]

Step 5: Answer.
\[ \boxed{1 + \log_e x} \]