![Q.18 d/dx}[ln(2x)]) is equal to (A) 1/(2x) (B) 1/x (C) 1/2 (D) x](data:image/svg+xml;base64,PHN2ZyB2aWV3Qm94PSIwIDAgMTkyMCAxMDgwIiB3aWR0aD0iMTkyMCIgaGVpZ2h0PSIxMDgwIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==)
Q.18 d/dx}[ln(2x)]) is equal to
- (A) 1/(2x)
- (B) 1/x
- (C) 1/2
- (D) x
The derivative of d/dx [ln(2x)] equals 1/x, making option (B) correct. This fundamental calculus result follows from the chain rule and logarithmic properties. Understanding why requires examining differentiation rules and evaluating each option.
Correct Answer
d/dx [ln(2x)] = 1/x
Apply the chain rule: let u = 2x, so ln(2x) = ln u.
Then
d/dx [ln u] = (1/u) ⋅ (du/dx) = (1/2x) ⋅ 2 = 1/x
Alternatively, use ln(2x) = ln 2 + ln x, where the derivative of constant ln 2 is 0, leaving 1/x.
Option Analysis
| Option | Expression | Correct for ln(2x)? | Explanation |
|---|---|---|---|
| (A) | 1/(2x) | No | This is the derivative of inner function 2x applied to ln x alone, ignoring chain rule multiplication by 2. |
| (B) | 1/x | Yes | Matches chain rule result: (1/2x) × 2 = 1/x. |
| (C) | 1/2 | No | Derivative of constant ln 2, but ignores variable x term. |
| (D) | x | No | Unrelated; resembles reciprocal of 1/x but incorrect form. |
Proof Using Log Properties
Rewrite ln(2x) = ln 2 + ln x.
Differentiate: d/dx [ln 2 + ln x] = 0 + 1/x = 1/x.
Both chain rule and log properties confirm the result consistently.
Common Mistakes
Students often pick (A) by forgetting the derivative of 2x is 2, which cancels the 2x denominator. Practice with similar forms like ln(kx) yields 1/x for any constant k. Verify by checking limits or plugging values, such as x=1 where slope matches 1/x behavior.
