Question

If $y=3^{x\log x},$ then $\frac{dy}{dx}=$

• A. $3^{x\log x}(1+\log x)\log 3$
• B. $3^{x\log x}(1+\log x)$
• C. $3^x(1+\log x)\log 3$
• D. $(1+\log x)\log 3$

Answer 1

The correct option is A $3^{x\log x}(1+\log x)\log 3$

We have,
$y=3^{x\log x}$
Differentiate it with respect to $x,$
$\frac{dy}{dx}=\frac{d}{dx}\left(3^{x\log x}\right)$
$=3^{x\log x}\times \log 3\times \frac{d}{dx}\left(x\log x\right)\text{[Using chain rule ]}$
$=3^{x\log x}\times \log 3\left(x\frac{d}{dx}\right(\log x)+\log x\frac{d}{dx}x)\text{[Using product rule ]}$
$=3^{x\log x}\times \log 3(\frac{x}{x}+\log x)$
$=3^{x\log x}(1+\log x)\log 3$