GATE BT 2021 • Q18
MCQ • 1 Mark
Section 1: Engineering Mathematics
Calculus
Subtopic: Fundamentals of Calculus
Question Statement:
The derivative d/dx [ln(2x)] with respect to x is equal to:
Options:
-
(A) 1 / (2x)
-
(B) 1 / x
-
(C) 1 / 2
-
(D) x
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2021 Q.18)**
**1. Governing Concept**:
- **Topic**: Engineering Mathematics & Differential Calculus (Chain Rule)
- Differentiating $f(x) = \ln(2x)$ with respect to $x$:
- Method 1 (Chain rule):
$\frac{d}{dx}[\ln(2x)] = \frac{1}{2x} \cdot \frac{d}{dx}(2x) = \frac{1}{2x} \cdot 2 = \frac{1}{x}$
- Method 2 (Logarithmic property):
$\ln(2x) = \ln 2 + \ln x \implies \frac{d}{dx}[\ln 2 + \ln x] = 0 + \frac{1}{x} = \frac{1}{x}$
**2. Option Analysis**:
- **Option (A) [1 / (2x)] is INCORRECT**: Forgets to multiply by the inner derivative 2.
- **Option (B) [1 / x] is CORRECT**: Exactly $\frac{1}{x}$.
- **Option (C) [1 / 2] is INCORRECT**: Derivative of linear function.
- **Option (D) [x] is INCORRECT**: Integration form.
**3. Final Answer**:
- **Correct Choice**: **B**
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