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

Question

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

✓

Answer

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

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 Differentiation→Differentiation — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{a}(\theta+\sin\theta)$ and $\text{y}=\text{a}(1-\cos\theta)$

Solve the following systems of linear equations by cramer's rule:

If y $ = \log\bigg[\text{x+}\sqrt{\text{x}^{2} + \text{a}^{2}}\bigg],\text {show that } (\text{x}^{2} + \text{a}^{2})\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} + \text{x}\frac{\text{dy}}{\text{dx}} = 0.$

Evaluate the following intregals: $\int\frac{4\text{x}^2+3}{(\text{x}^2+2)(\text{x}^2+3)(\text{x}^2+4)}\ \text{dx}$

One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5kg of flour and 1kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

Find the equation of the curve through the point (1, 0) if the slope of the tangent to the curve at any point (x, y) is $\frac{\text{y}-1}{\text{x}^2+\text{x}}.$

Evaluate the following integrals:
$\int\frac{\text{x}^2+9}{\text{x}^4+81}\ \text{dx}$

Solve the following differential equation
$\text{xy}(\text{y}+1)\text{dy}=(\text{x}^2+1)\text{dx}$

Prove that:
$\begin{vmatrix}1&1+\text{p}&1+\text{p}+\text{q}\\2&3+2\text{p}&4+3\text{p}+2\text{p}\\3&6+3\text{p}&10+6\text{p}+3\text{q}\end{vmatrix}=1$

Solve the following system of equations by matrix method:$\frac{2}{\text{x}}-\frac{3}{\text{y}}+\frac{3}{\text{z}}=10$
$\frac{1}{\text{x}}+\frac{1}{\text{y}}+\frac{1}{\text{z}}=10$
$\frac{3}{\text{x}}-\frac{1}{\text{y}}+\frac{2}{\text{z}}=13$