If \(y = e^{nx}\), then nth derivative of y is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(n^ny\)
By the chain rule, each differentiation of \(e^{nx}\) brings out one more factor n:
\( \displaystyle \frac{dy}{dx} = ne^{nx} \)
\( \displaystyle \frac{d^2y}{dx^2} = n^2e^{nx} \)
\( \displaystyle \frac{d^3y}{dx^3} = n^3e^{nx} \)
Continuing, after n differentiations:
\( \displaystyle \frac{d^ny}{dx^n} = n^ne^{nx} = n^ny \)
Why the other options are incorrect
Option (1) \(e^{nx}\) — Incorrect. This leaves out the factor n that each differentiation brings out.
Option (2) \(n^2e^{nx}\) — Incorrect. This is only the second derivative.
Option (3) \(ny\) — Incorrect. This is only the first derivative.