Differentiate the following functions with respect to x: e^x x^2 - Vidyadip

Question

Differentiate the following functions with respect to x:
$\frac{\text{e}^\text{x}\log\text{x}}{\text{x}^2}$

✓

Answer

Let $\text{y}=\frac{\text{e}^\text{x}\log\text{x}}{\text{x}^2}$
Differentiate with respect to x we get,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}^2\frac{\text{d}}{\text{dx}}(\text{e}^\text{x}\log\text{x})-(\text{e}^\text{x}\log\text{x})\frac{\text{d}}{\text{dx}}\text{x}^2}{\big(\text{x}^2\big)^2}$
[Using quotient rule]
$=\frac{\text{x}^2\Big\{\text{e}^\text{x}\frac{\text{d}}{\text{dx}}(\log\text{x})+\log\text{x}\frac{\text{d}}{\text{dx}}(\text{e}^\text{x})\Big\}-\text{e}^\text{x}\log\text{x}\times2\text{x}}{\text{x}^4}$
[Using product rule]
$=\frac{\text{x}^2\Big[\frac{\text{e}^\text{x}}{\text{x}}+\text{e}^\text{x}\log\text{x}\Big]-2\text{xe}^\text{x}\log\text{x}}{\text{x}^4}$
$=\frac{\frac{\text{x}^2\text{e}^\text{z}(1+\text{x}\log\text{x})}{\text{x}}-2\text{xe}^\text{z}\log\text{x}}{\text{x}^4}$
$=\frac{\text{xe}^\text{x}[1+\text{x}\log\text{x}-2\log\text{x}]}{\text{x}^4}$
$=\frac{\text{xe}^\text{x}}{\text{x}^3}\Big[\frac{1}{\text{x}}+\frac{\text{x}\log\text{x}}{\text{x}}-\frac{2\log\text{x}}{\text{x}}\Big]$
$=\text{e}^\text{x}\text{x}^{-2}\Big[\frac{1}{\text{x}}+\log\text{x}-\frac{2}{\text{x}}\log\text{x}\Big]$
So,
$\frac{\text{d}}{\text{dx}}\Big[\frac{\text{e}^\text{x}\log\text{x}}{\text{x}^2}\Big]=\text{e}^\text{x}\text{x}^{-2}\Big[\frac{1}{\text{x}}+\log\text{x}-\frac{2}{\text{x}}\log\text{x}\Big]$

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 "5 Marks Questions" 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

Evaluate the following integrals:
$\int\frac{1}{\text{x}^3}\sin(\log\text{x})\text{dx}$

Make a sketch of the recion ${(x, y) : 0 < y < x^2 + 3 : 0 < y < 2x + 3 : 0 < x < 3}$ and find using interation.

Find the point on the curve $y^2= 4x$ which is nearest to the point $(2, -8).$

If A and B are two independent events such that $\text{P}(\overline{\text{A}}\cap\text{B})=\frac{2}{15}$ and $\text{P}(\text{A}\cap\overline{\text{B}})=\frac{1}{6}$, then find P(B).

In order to supplement daily diet, a person wishes to take $X$ and $Y$ tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in $X$ and $Y$ are given as below:

Tablets	Iron	Calcium	Vitamin
$X$	$6$	$3$	$2$
$Y$	$2$	$3$	$4$

The person needs to supplement at least $18$ milligrams of iron, $21$ milligrams of calcium and $16$ milligrams of vitamins. The price of each tablet of $X$ and $Y$ is $₹ 2$ and $₹1$ respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.

An urn contains 5 red and 2 blcak balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X? Is X a random variable? If yes, then find the mean and variance of X.

Let R be a relation on the set N given by R = {(a, b): a = b - 2, b > 6}. Then,

(2, 4) ∈ R
(3, 8) ∈ R
(6, 8) ∈ R
(8, 7) ∈ R

Show that the four points A, B, C and D with the position vectors $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ and $\vec{\text{d}}$ respectively are coplanar if and only if $3\vec{\text{a}}-2\vec{\text{b}}+\vec{\text{c}}-2\vec{\text{d}}=\vec0$.

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Find the angle of intersecting of the following curves:
$\text{y}=\text{x}^2\text{ and }\text{x}^2+\text{y}^2=20$