Question

If $y=2^{x\ln x^2},$ then derivative of $y$ with respect to $x$ is

[2 marks]

• A. $\left(\ln 2\right)(1+\ln x)2^{x\ln x^2}$
• B. $\left(2\ln 2\right)(1+\ln x)2^{x\ln x^2}$
• C. $\left(2\ln 2\right)(1-\ln x)2^{x\ln x^2}$
• D. $\left(2\ln 2\right)(1+\ln x)2^{x\ln x}$

Answer 1

The correct option is B $\left(2\ln 2\right)(1+\ln x)2^{x\ln x^2}$

Given : $y=2^{x\ln x^2}$
Taking $\ln$ on both sides, we have
$\ln y=x\ln x^2\ln 2$
$\Rightarrow \ln y=\left(2\ln 2\right)x\ln x$
Differentiating with respect to $x,$ we get
$\frac{1}{y}\frac{dy}{dx}=\left(2\ln 2\right)(x\cdot \frac{1}{x}+1\cdot \ln x)$
$\Rightarrow \frac{dy}{dx}=\left(2\ln 2\right)(1+\ln x)y$
$\therefore \frac{dy}{dx}=\left(2\ln 2\right)(1+\ln x)2^{x\ln x^2}$