If y = x^ x^2 then dy dx = - Vidyadip

MCQ

If $y ={x^{{x^2}}}$ then $\frac{{dy}}{{dx}}=$

A
$2 \, ln \, x .{x^{{x^2}}}$
B
$(2 \, ln \, x + 1).{x^{{x^2}}}$
C
$(2 \, ln \, x + 1).{x^{{x^2+2}}}$
✓
${x^{{x^2}}}. \, ln \, ex^2$

✓

Answer

Correct option: D.

${x^{{x^2}}}. \, ln \, ex^2$

d
We have, $y=x^{x}$

Taking log on both the sides, we get $\log y=x \log x$

On differentiating w.r.t. $x$, we get $\frac{1}{y} \frac{d y}{d x}=\frac{x}{x}+\log x$

$\Rightarrow \frac{d y}{d x}=y+y \log x$

$\Rightarrow \frac{d y}{d x}=x^{x}(1+\log x) \ldots\left(\because y=x^{x}\right)$

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $A$ and $B$ be two finite sets having $m$ and $n$ elements respectively such that $m \le n.\,$ A mapping is selected at random from the set of all mappings from $A$ to $B$. The probability that the mapping selected is an injection is

The probability of men getting a certain disease is $\frac{1}{2}$ and that of women getting the same disease is $\frac{1}{5}$. The blood test that identifies the disease gives the correct result with probability $\frac{4}{5}$. Suppose a person is chosen at random from a group of $30$ males and $20$ females, and the blood test of that person is found to be positive. What is the probability that the chosen person is a man?

The value of $\sum\limits_{k = 1}^\infty {\frac{{3{k^2} + 3k + 1}}{{{{\left( {{k^2} + k} \right)}^3}}}} $ is equal to

If $\alpha $, $\beta $ and $\gamma $ are three consecutive terms of a non-constant $G.P.$ such that the equations $\alpha x^2 + 2\beta x + \gamma = 0$ and $x^2 + x -1 = 0$ have a common root, then $\alpha(\beta + \gamma )$ is equal to

The position of the point $(4, -3)$ with respect to the ellipse $2{x^2} + 5{y^2} = 20$ is

Let $D_1 =$ $\left| {\,\begin{array}{*{20}{c}}a&b&{a + b}\\c&d&{c + d}\\a&b&{a - b}\end{array}\,} \right|$ and $D_2 =$ $\left| {\,\begin{array}{*{20}{c}}a&c&{a + c}\\b&d&{b + d}\\a&c&{a + b + c}\end{array}\,} \right|$ then the value of $\frac{{{D_1}}}{{{D_2}}}$ where $b \ne 0$ and $ad \ne bc$, is

If the sum of three numbers of a arithmetic sequence is $15$ and the sum of their squares is $83$, then the numbers are

The sum of all those terms which are rational numbers in the expansion of $\left(2^{1 / 3}+3^{1 / 4}\right)^{12}$ is:

If $A, B, C$ be three sets such that $A \cup B = A \cup C$ and $A \cap B = A \cap C$, then

A function $y = f (x) $ satisfies $(x + 1) . f ' (x) -2 (x^2 + x) f (x) = \frac{{{e^{{x^2}}}}}{{(x + 1)}}, \forall x >-1$ . If $f$ $(0)$ $=$ $5$ , then $f$ $(x)$ is