The derivative of x^ 2 x w.r.t. x is - Vidyadip

MCQ

The derivative of $x^{2 x}$ w.r.t. $x$ is

A
$x^{2 x-1}$
B
$2 x^{2 x} \log x$
C
$2 x^{2 x}(1+\log x)$
D
$2 x^{2 x}(1-\log x)$

✓

Answer

Let $y=x^{2 x}$
Taking log on both sides, we get
$
\log y=2 x \log x
$
Differentiating both sides w.r.t. $x$, we get
$
\begin{aligned}
& \frac{1}{y} \frac{d y}{d x}=2\left\{x \cdot \frac{1}{x}+\log x \cdot 1\right\} \\
\Rightarrow & \frac{d y}{d x}=2 y\{1+\log x\}=2 x^{2 x}(1+\log x)
\end{aligned}
$

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