Find dy dx y=x^ x +x^1 x - Vidyadip

Question

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{x}^{\text{x}}+\text{x}^\frac{1}{\text{x}}$

✓

Answer

Here,
$\text{y}=\text{x}^{\text{x}}+\text{x}^\frac{1}{\text{x}}$
$=\text{e}^{\log\text{x}^\text{x}}+\text{e}^{\log\text{x}^\frac{1}{\text{x}}}$
$\text{y}=\text{e}^{\text{x}\log\text{x}}+\text{e}^{\big(\frac{1}{\text{x}}\log\text{x}\big)}$
$\big[\text{Since, e}^{\log\text{a}}=\text{a},\log\text{a}^\text{b}=\text{b}\log\text{a}\big]$
Differentiating it with respect to x using chain rule and product rule,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\text{e}^{\text{x}\log\text{x}}\big)+\frac{\text{d}}{\text{dx}}\Big(\text{e}^{\frac{1}{\text{x}}\log\text{x}}\Big)$
$=\text{e}^{\text{x}\log\text{x}}+\frac{\text{d}}{\text{dx}}(\text{x}\log\text{x})+\text{e}^{\frac{1}{\text{x}}\log\text{x}}\frac{\text{d}}{\text{dx}}\Big(\frac{1}{\text{x}}\log\text{x}\Big)$
$=\text{e}^{\text{x}\log\text{x}}\Big[\text{x}\frac{\text{d}}{\text{dx}}(\log\text{x})+\log\text{x}\frac{\text{d}}{\text{dx}}(\text{x})\Big] \\ +\text{e}^{\log\text{x}^\frac{1}{\text{x}}}\Big[\frac{1}{\text{x}}\frac{\text{d}}{\text{dx}}(\log\text{x})+\log\text{x}\frac{\text{d}}{\text{dx}}\big(\frac{1}{\text{x}}\big)\Big]$
$=\text{x}^{\text{x}}\Big[\text{x}\Big(\frac{1}{\text{x}}\Big)+\log\text{x}(1)\Big] \\ +\text{x}^\frac{1}{\text{x}}\Big[\Big(\frac{1}{\text{x}}\Big)\Big(\frac{1}{\text{x}}\Big)+\log\text{x}\Big(-\frac{1}{\text{x}^2}\Big)\Big]$
$=\text{x}^\text{x}[1+\log\text{x}]+\text{x}^{\frac{1}{\text{x}}}\Big(\frac{1}{\text{x}^2}-\frac{1}{\text{x}^2}\log\text{x}\Big)$
$\frac{\text{dy}}{\text{dx}}=\text{x}^\text{x}[1+\log\text{x}]+\text{x}^{\frac{1}{\text{x}}}\frac{(1-\log\text{x})}{\text{x}^2}$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Differentiate $\tan^{-1}\Big(\frac{\text{x}}{\sqrt{1-\text{x}^2}}\Big)$ with respect to $\sin^{-1}\Big(2\text{x}\sqrt{1-\text{x}^2}\Big),$ if $-\frac{1}{\sqrt{2}}<\text{x}<\frac{1}{\sqrt{2}}$

In a group of $400$ people, $160$ are smokers and non$-$vegetarian, $100$ are smokers and vegetarian and the remaining are non$-$smokers and vegetarian. The probabilities of getting a special chest disease are $35\%, 20\%$ and $10\%$ respectively. A person is chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non$-$vegetarian?

If $\text{x}=\text{a}(\cos\text{t}+\log\tan\frac{\text{t}}{2})\ \text{and}\ \text{y}=\text{a}(\sin\text{t}),$evaluate $\frac{\text{d}^2\text{y}}{\text{dx}^2}\ \text{at}\ \text{t}=\frac{\pi}{3}.$

Evalute the following integrals:
$\int\frac{\cos\text{x}}{2+3\sin\text{x}}\text{dx}$

Evaluate the following integrals: $\int_{0}^\limits{1}\frac{1}{1+2\text{x}+2\text{x}^2+2\text{x}^3+\text{x}^4}\text{ dx}$

Evaluate the follwing intregals:
$\int\frac{1}{\text{x}(\text{x}^4-1)}\ \text{dx}$

Evaluate the following integrals as limit of sum:
$\int\limits^{\frac{\pi}{2}}_{0}\sin\text{x dx}$

Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}0&2&6\\1&5&0\\3&7&1 \end{vmatrix}$

Using elementary transformations, find the inverse of the matrix $\text{A} = \begin{bmatrix} 8 & 4 & 3 \\ 2 & 1 & 1 \\ 1 & 2 & 2 \end{bmatrix}$ and use it to solve the following system of linear equations:
$\text{8x + 4y +3z = 19}$
$\text{2x + y + z = 5} $
$\text{x + 2y + 2z = 7}$

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?