If y = x^x, find dy dx at x=e - Vidyadip

Question

If $y = x^x,$ find $\frac{\text{dy}}{\text{dx}}\text{at x}=\text{e}$

✓

Answer

We have, $y = x^x .....(i)$
Taking log on both sides,
$\log\text{y}=\log\text{x}^\text{x}$
$\Rightarrow\log\text{y}=\text{x}\log\text{x}$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\text{x}\frac{\text{d}}{\text{dx}}(\log\text{x})+\log\text{x}\frac{\text{d}}{\text{dx}}(\text{x})$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\text{x}\Big(\frac{1}{\text{x}}\Big)+\log\text{x}(1)$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=1+\log\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}(1+\log\text{x})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{x}^\text{x}(1+\log\text{x})$
$[$Using equation $(i)]$
Putting $x = e$, we get,
$\frac{\text{dy}}{\text{dx}}=\text{e}^\text{e}(1+\log_\text{e}\text{e})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{e}^\text{e}(1+1)\big[\because\log_\text{e}\text{e}=1\big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=2\text{e}^\text{e}$

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 "3 Marks Question" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Write the vector equation of the line passing through the point (1, -2, -3) and normal to the plane $\vec{\text{r}}.(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}})=5.$

$\text{Solve for x:}$
$2\tan^{-1}(\cos x) = \tan^{-1}(2\text{cosec x)} $

Show that the relation R on defined as $\text{R}=\big\{(\text{a},\text{b}):\text{a}\leq\text{b}\big\},$ is reflexive, and transitive but not symmetric.

Evaluate the following integrals:$\int\frac{\text{x}+7}{3\text{x}^2+25\text{x}+28}\text{ dx}$

Find the probability distribution of white balls drawn in a random draw of 3 balls without replacement, from a bag containing 4 white and 6 red balls.

Find the mean and standard deviation of the following probability distributions:

$x_i$	$2$	$3$	$4$
$p_i$	$0.2$	$0.5$	$0.3$

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (-1, 0, 0), (0, 1, 2), respectively, then find $\angle\text{ABC}.[\angle\text{ABC is the angle between the vectors}\ \overrightarrow{\text{BA}}\ \text{and} \ \overrightarrow{\text{BC}}].$

If $A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]$ show that $\mathrm{A}^2-5 \mathrm{~A}+7 \mathrm{I}=0$

The two adjacent sides of a parallelogram are $2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}\ \text{and}\ \hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}}.$ Find the unit vector parallel to its diagonal. Also, find its area.

A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared at least once?