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}
$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Continuity and Differentiability→Continuity and Differentiability — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\int {{e^{{x^2}}}\left( {2 - \frac{1}{{{x^2}}}} \right)dx = {e^{{x^2}}}f(x) + C} $ and $f\left( {\frac{1}{2}} \right) = 2$ , then $f(1)$ is equal to (where $C$ is an arbitrary constant)

One vertex of a rectangular parallelopiped is at the origin $O$ and the lengths of its edges along $x , y$ and $Z$ axes are $3,4$ and $5$ units respectively. Let $P$ be the vertex $(3,4,5)$. Then the shortest distance between the diagonal $OP$ and an edge parallel to $Z$ axis, not passing through $O$ or $P$ is:

The length of the perpendicular from the point $(2, -1, 4)$ on the straight line, $\frac{{x + 3}}{{10}} = \frac{{y - 2}}{{ - 7}} = \frac{z}{1}$ is

If $\cot^{-1}(\sqrt{\cos\alpha})-\tan^{-1}(\sqrt{\cos\alpha})=\text{x},$ then $\sin\text{x}$ is equal to:

$\tan^2\Big(\frac{\alpha}{2}\Big)$
$\cot^2\Big(\frac{\alpha}{2}\Big)$
$\tan\alpha$
$\cot\Big(\frac{\alpha}{2}\Big)$

Choose the correct answer from the given four options.
The value of $\cot\Big[\cos^{-1}\Big(\frac{7}{25}\Big)\Big]$ is:

$\frac{25}{24}$
$\frac{25}{7}$
$\frac{24}{25}$
$\frac{7}{24}$

Choose the correct answer from the given four options.

For any two matrices A and B, we have:

$\text{AB}=\text{BA}$
$\text{AB}\neq\text{BA}$
$\text{AB}=\text{O}$
None of the above.

Objective of LPP is:

Mark the correct alternative in the following question for the binary operation * on Z defined by a * b = a + b + 1, the identity element is:

0
-1
1
2

If $f:\left\{ {1,2,3,4} \right\} \to \left\{ {1,2,3,4} \right\}$ and $y=f(x)$ be a function such that $\left| {f\left( \alpha \right) - \alpha } \right| \leqslant 1$,for $\alpha \in \left\{ {1,2,3,4} \right\}$ then total number of such functions are

${d \over {dx}}\left[ {\log \sqrt {\sin \sqrt {{e^x}} } } \right]=$