Differentiate ^ -1 (2 x+3 x 13 ) w.r. to x - Vidyadip

Question

Differentiate $\sin ^{-1}\left(\frac{2 \cos x+3 \sin x}{\sqrt{13}}\right)$ w.r. to $x$

✓

Answer

$ \text { Let } y =\sin ^{-1}\left(\frac{2 \cos x+3 \sin x}{\sqrt{13}}\right)$
$=\sin ^{-1}\left(\frac{2 \cos x}{\sqrt{13}}+\frac{3 \sin x}{\sqrt{13}}\right) $
Put $\frac{2}{\sqrt{13}}=\sin t$ and $\frac{3}{\sqrt{13}}=\cos t$
Also, $\sin ^2 t +\cos ^2 t =\frac{4}{13}+\frac{9}{13}=1$
and $\tan t =\frac{2}{3}$
$\therefore t =\tan ^{-1}\left(\frac{2}{3}\right)$
$\therefore y=\sin ^{-1}(\sin t \cdot \cos x+\cos t \cdot \sin x)$
$=\sin ^{-1}[\sin (t+x)]$
$= t + x$
$=\tan ^{-1}\left(\frac{2}{3}\right)+x$
Differentiating w. r. t. x, we get
$ \frac{ d y}{ d x}=\frac{ d }{ d x}\left[\tan ^{-1}\left(\frac{2}{3}\right)+x\right]$
$=0+1$
$=1 $

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 "Solve the Following Question.(3 Marks)" from Question Bank [2022]→Question Bank [2022] — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Reduce each of the following differential equations to the separted variable form and hence find the general solution.$\frac{d y}{d x}=(4 x+y+1)^2$

Evaluate the following integrals:$\int^\limits{\frac{\pi}{2}}_{\frac{-\pi}{2}}\big\{\sin|\text{x}|+\cos|\text{x}|\big\}\text{dx}$

Find k ifThe slope of one of the lines given by $3 x^2+4 x y+k y^2=0$ is three times the other.

For what value of k is the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin5\text{x}}{3\text{x}}, &\text{if}\text{ x}\neq0\\\text{k}, &\text{if}\text{ x}=0\end{cases}$ is continuous at x = 0?

Obtain the differential euqation by eliminating the arbitrary constants from the following :
$y=c_1 e^{3 x}+c_2 e^{2 x}$

Evaluate the following integrals:
$\int\sec^6\text{x }\tan\text{x}\text{ dx}$

Show that $\text{f}(\text{x})=\cos^2\text{x}$ is a decreasing function on $\Big(0,\frac{\pi}{2}\Big).$

The probability distribution of discrete random variable $X$ is as follows:

$X =x$	1	2	3	4	5	6
$P ( X =x)$	$k$	$2k$	$3k$	$4k$	$5k$	$6k$

Find (1) the value of $k$
(ii) $P (2< X <4)$
(iii) $P ( X \geq 3$ )

Find the equation of the plane with intercept 3 on the y-axis and parallel to the ZOX plane.

The probability of a man hitting a target is 1/4. If he fires 7 times, what is the probability of his hitting the target at least twice?