If \( y=\log_e\sin(e^x+5x+500) \) then \( \displaystyle\frac{dy}{dx} \) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( (e^x+5)\cot(e^x+5x+500) \)
By the chain rule, working from the outside in:
\( \displaystyle \frac{dy}{dx}=\frac{1}{\sin(e^x+5x+500)}\cdot\cos(e^x+5x+500)\cdot(e^x+5) \)
\( =(e^x+5)\cot(e^x+5x+500) \)
Why the other options are incorrect
Option (2) \( -(e^x+5)\cot(e^x+5x+500) \) — Incorrect. The derivative of sin u is cos u, with no minus sign. The minus sign would appear only if the function were cos u.
Option (3) \( \cot(e^x+5x+500) \) — Incorrect. This misses the derivative of the inner function, \( e^x+5 \).
Option (4) \( 5e^x\sin(e^x+5x+500) \) — Incorrect. The derivative has cot, not sin, and the factor is \( e^x+5 \), not \( 5e^x \).