\( \displaystyle\frac{d}{dx}\left(\displaystyle\frac{4^x}{x^4}\right) \) equals
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{4^x}{x^4}\left[\log 4-\displaystyle\frac{4}{x}\right] \)
Written as a product, the function is \( 4^x\cdot x^{-4} \), and \( \frac{d}{dx}a^x=a^x\log a \).
\( \displaystyle \frac{d}{dx}\left(4^x x^{-4}\right)=4^x\log 4\cdot x^{-4}+4^x\cdot(-4x^{-5}) \)
Taking out \( \frac{4^x}{x^4} \):
\( \displaystyle =\frac{4^x}{x^4}\left[\log 4-\frac{4}{x}\right] \)
Shortcut: Logarithmic differentiation gives the bracket directly: with \( y=\frac{4^x}{x^4} \), \( \log y=x\log 4-4\log x \), so
\( \displaystyle \frac{1}{y}\frac{dy}{dx}=\log 4-\frac{4}{x} \)
Why the other options are incorrect
Option (2) \( \displaystyle\frac{4^x}{x^4}\left[\log 4+\displaystyle\frac{4}{x}\right] \) — Incorrect. The sign of the second term is wrong: the derivative of \( x^{-4} \) is \( -4x^{-5} \).
Option (3) \( \displaystyle\frac{4^x}{x^4}[\log 4-4x] \) — Incorrect. The derivative of \( x^{-4} \) is \( -4x^{-5} \), which gives \( \frac{4}{x} \) inside the bracket, not 4x.
Option (4) \( \displaystyle\frac{4^x}{x^4}[\log 4+4x] \) — Incorrect. Both the sign and the term are wrong; the bracket should contain \( -\frac{4}{x} \).