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

Question

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

✓

Answer

Let $\text{y}=\text{e}^{\text{x}\log\text{x}}$
$\Rightarrow\ \text{y}=\text{e}^{\log\text{x}^\text{x}} \ \big[\text{Since}, \log\text{a}^\text{b}=\text{b}\log\text{a}\big]$
$\Rightarrow\text{y}=\text{x}^{\text{x}}\ .....(\text{i})\ \big[\text{Since, e}^{\log\text{a}}=\text{a}\big]$
Taking log on both the sides,
$\log\text{y}=\log\text{x}^\text{x}$
$\log\text{y}=\text{x}\log\text{x}$
Differentiating with respect to x, using product rule,
$\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\text{x}\frac{\text{d}}{\text{dx}}(\log\text{x})+\log\text{x}\frac{\text{d}}{\text{dx}}(\text{x})$
$=\text{x}\Big(\frac{1}{\text{x}}\Big)+\log\text{x}(1)$
$\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=1+\log\text{x}$
$\frac{\text{dy}}{\text{dx}}=\text{y}[1+\log\text{x}]$
$\frac{\text{dy}}{\text{dx}}=\text{x}^\text{x}(1+\log\text{x})$
[Using equation (i)]

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 Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Minimise and Maximise Z = x + 2y
subject to $\text{x}+2\text{y}\geq100,\ 2\text{x}-\text{y}\leq0,\ 2\text{x}+ \text{y}\leq200;\ \text{x},\ \text{y}\geq0.$

A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.

If $\vec{\text{p}}=5\hat{\text{i}}+\lambda\hat{\text{j}}-3\hat{\text{k}}$ and $\vec{\text{q}}=\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}},$ then find the value of $\lambda,$ so that $\vec{\text{p}}+\vec{\text{q}}$ and $\vec{\text{p}}-\vec{\text{q}}$ are perpendicular vectora.

A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.

Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is $\frac{1}{4}$ and that to Tarun's rejection is $\frac{2}{3}$. Find the probability that at least one of them will be selected.

Solve the following initial value problems:
$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y + x}\sin\Big(\frac{\text{y}}{\text{x}}\Big)=0,\text{y}(2)=\text{x}$

Using properties of definite integrals, evaluate:
$\int\limits^{\pi/4}_{0}\text{log (1 + tan x) dx}$.

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹ 1,600. School B wants to spend ₹ 2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹ 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

Differentiate w.r.t. x the function in Exercise:
$(\sin\text{x}-\cos\text{x)}^{(\sin\text{x}-\cos\text{x})},\ \frac{\pi}{4}<\text{x}<\frac{3\pi}{4}$