Differentiate the following function w.r.t. x:X^ x + ( x)^ x - Vidyadip

Question

Differentiate the following function w.r.t. x:$X^{\sin x} + (\sin x)^{\cos x} $

✓

Answer

$\text{Let x}^{\sin x} = \text{u, and} ( \sin \text{x)}^{\cos \text{x}} = \text{v} \therefore \text{y = u + v} \Rightarrow \frac{\text {dy}}{\text{dx}} = \frac{\text{du}}{\text{dx}} + \frac{\text{dv}}{\text{dx}}$

$\text{Getting} \frac{\text{du}}{\text{dx}} = \text{x}^{\sin{\text{x}}} \bigg[\frac{\sin\text{x}}{\text{x}} + \log \text{x} . \cos{\text{x}} \bigg]$

$\text{and} \frac{\text{dv}}{\text{dx}} = ( \sin \text{x})^{\cos\text{x}}[\cos\text{x} .\cot\text{x} - \sin\text{x} \log \sin \text{x}]$

$\therefore \frac{\text{dy}}{\text{dx}} = \text{x}^{\sin \text{x}} \bigg[ \frac{\sin \text{x}}{\text{x}} + \log\text{x}.\cos\text{x}\bigg] + (\sin\text{x}) ^{\cos{\text{x}}} [\cos\text{x}. \cot\text{x} -\sin\text{x} \log\sin\text{x}]$

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: $\int\limits_0^{\frac{\pi}{4}}\frac{\sin\text{x}+\cos\text{x}}{16+9\sin2\text{x}}$

Coloured balls are distributed in $3$ bags as shown in the following table :

Colour of balls
Bag	Black	White	Red
$I$	$1$	$2$	$3$
$1$	$2$	$4$	$1$
$III$	$4$	$5$	$3$

A bag is selected at random and then two balls are randomly drawn from the selected bag. The colours of the balls are black and red. What is the probability that ball drawn is from bag I?

Find the equation of the plane which is parallel to 2x - 3y + z = 0 and which passes through (1, -1, 2).

Solve the following differential equation:
$(\text{x}+2\text{y})\text{dx}-(2\text{x}-\text{y})\text{dy}=0$

Show that $\int_0^a {f(x)g(x)dx = 2\int_0^a {f(x)dx} } $. If f and g are defined, $f(x) = f(a - x)$ and $g(x) + g(a - x) = 4$

If A = $\begin{bmatrix}1&0&2\\0&2&1\\2&0&3\end{bmatrix}$, prove that $A^3 - 6A^2 + 7A + 2I = 0$.

Find the maximum and the minimum values, if any, without using derivaives of the following functions:
$f(x) = |x + 2| + 2$ on $R$.

Find the matrix A satisfying the matrix equation:$\begin{bmatrix}2&1\\3&2\end{bmatrix}\text{A}\begin{bmatrix}-3&2\\5&-3\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}.$

A manufacturer has three machines installed in his factory. machines I and II are capable of being operated for at most 12 hours whereas Machine III must operate at least for 5 hours a day. He produces only two items, each requiring the use of three machines. The number of hours required for producing one unit each of the items on the three machines is given in the following table:

Item	Number of hours required by the machine
	I	II	III
A	1	2	1
B	2	1	$\frac{5}{4}$

He makes a profit of Rs. 6.00 on item A and Rs. 4.00 on item B. Assuming that he can sell all that he produces, how many of each item should he produces so as to maximize his profit? Determine his maximum profit. Formulate this LPP mathematically and then solve it.

Let L be the set of all lines in xy plane and R be the relation in L. define as $s R =\left\{\left( L _1, L_2\right): L _1 \| L _2\right\}$ Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4