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: Write the given function.
\[ y=x\log_e(x) \]

Step 2: Identify the differentiation rule.
The function is a product of two functions:
\[ u=x,\qquad v=\log_e(x) \] So, we use the product rule.

Step 3: Recall the product rule formula.
If
\[ y=uv, \] then
\[ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx} \]

Step 4: Differentiate each factor.
\[ \frac{du}{dx}=1 \] and
\[ \frac{dv}{dx}=\frac{1}{x} \]

Step 5: Substitute into the product rule.
\[ \frac{dy}{dx} = x\left(\frac{1}{x}\right) + \log_e(x)(1) \]

Step 6: Simplify the expression.
\[ \frac{dy}{dx} = 1+\log_e(x) \]

Step 7: Final conclusion.
Hence, the derivative of \(x\log_e(x)\) is
\[ \boxed{1+\log_e(x)} \]
Therefore, the correct answer is option (D).