If \(y = (3x - 2)^4\), then the value of \(\displaystyle\frac{d^2y}{dx^2}\) at x = 1 is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 108
By the chain rule, every differentiation of \((3x - 2)^n\) brings down the power and a factor 3 (the derivative of \(3x - 2\)):
\( \displaystyle \frac{dy}{dx} = 4(3x - 2)^3 \times 3 = 12(3x - 2)^3 \)
\( \displaystyle \frac{d^2y}{dx^2} = 36(3x - 2)^2 \times 3 = 108(3x - 2)^2 \)
At x = 1, \(3x - 2 = 1\):
\( \displaystyle \frac{d^2y}{dx^2} = 108 \times 1 = 108 \)
Why the other options are incorrect
Option (1) 36 — Incorrect. The factor 3 from the inner function has been left out in the second differentiation.
Option (3) 12 — Incorrect. This is the first derivative at x = 1, \(12 \times 1^3\).
Option (4) 324 — Incorrect. An extra factor 3 has been used; there are only two differentiations, so only two factors of 3.