The derivative of \( x^{5x} \) with respect to \( x \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 5.x^{5x}(1+\log_e x) \)
Both the base and the power contain x, so neither the power rule nor the exponential rule applies; logarithmic differentiation is used. Let \( y=x^{5x} \):
\( \displaystyle \log y=5x\log x \)
\( \displaystyle \frac{1}{y}\frac{dy}{dx}=5\log x+5x\cdot\frac{1}{x} \)
\( \displaystyle \frac{1}{y}\frac{dy}{dx}=5(1+\log x) \)
\( \displaystyle \frac{dy}{dx}=5x^{5x}(1+\log_e x) \)
Why the other options are incorrect
Option (1) \( x^{5x-1} \) — Incorrect. This treats \( x^{5x} \) like \( x^n \) with a constant power (and even then the factor n is missing); the power here varies with x.
Option (2) \( 5x^{5x}.\log_e 5 \) — Incorrect. This treats \( x^{5x} \) like \( a^x \) with a constant base; the base here varies with x.
Option (4) \( 5.x^x(1-\log_e x) \) — Incorrect. Both the power of x (x instead of 5x) and the sign in the bracket are wrong.