Differentiation of a Function

Q. Differentiate ${\sin }^{-1}\left\{\frac{2^{x+1}·3^x}{1+{\left(36\right)}^x}\right\}$ with respect to x.

Q. $\int {\sin }^{7}xdx$

Q. If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find $\frac{d^{2}y}{d{x}^2}\mathrm{at}t=\frac{\mathrm{\pi }}{2}$.

Q. If $x=3\sin t-\sin 3t,y=3\cos t-\cos 3t\mathrm{find}\frac{dy}{dx}\mathrm{at}t=\frac{\mathrm{\pi }}{3}$

Q. derivative of 1-tanx/1+tanx

Q. If A = $\left[\begin{array}{cc}\cos 2\mathrm{\theta }& \sin 2\mathrm{\theta }\\ -\sin 2\mathrm{\theta }& \cos 2\mathrm{\theta }\end{array}\right]$, find A².

Q. If $\sin y=x\sin \left(a+y\right),\mathrm{then}\frac{dy}{dx}\mathrm{is}$

(a) $\frac{\sin a}{\sin a{\sin }^2\left(a+y\right)}$

(b) $\frac{{\sin }^2\left(a+y\right)}{\sin a}$

(c) $\sin a{\sin }^2\left(a+y\right)$

(d) $\frac{{\sin }^2\left(a-y\right)}{\sin a}$

Q. If , prove that

Q. Prove the following results:

(i) ${\tan }^{-1}\frac{1}{7}+{\tan }^{-1}\frac{1}{13}={\tan }^{-1}\frac{2}{9}$
(ii) ${\sin }^{-1}\frac{12}{13}+{\cos }^{-1}\frac{4}{5}+{\tan }^{-1}\frac{63}{16}=\mathrm{\pi }$
(iii) ${\tan }^{-1}\frac{1}{4}+{\tan }^{-1}\frac{2}{9}={\sin }^{-1}\frac{1}{\sqrt{5}}$

Q. Let $f:R\to Rsatisfyf(x+y)=2^{x}f\left(y\right)+4^{y}f\left(x\right)$, $\forall x,y\in R\text{. If}f\left(2\right)=3\text{, then}14\cdot \frac{f^{\prime }\left(4\right)}{f^{\prime }\left(2\right)}\text{is equal to.}$

Q. If $y=\left\{{\log }_{\cos x}\sin x\right\}{\left\{{\log }_{\sin x}\cos x\right\}}^{-1}+{\sin }^{-1}\left(\frac{2x}{1+x^2}\right),\mathrm{find}\frac{dy}{dx}atx=\frac{\pi }{4}$.

Q. $\mathrm{If}x=\cos t+\log \tan \frac{t}{2},y=\sin t,\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\frac{d^{2}y}{d{t}^2}\mathrm{and}\frac{d^{2}y}{d{x}^2}\mathrm{at}t=\frac{\mathrm{\pi }}{4}.$

Q. $\sin x=\frac{2t}{1+t^2},\tan y=\frac{2t}{1-t^2},\mathrm{find}\frac{dy}{dx}$

Q. Differentiate

$\sqrt{\frac{1+\sin x}{1-\sin x}}$

Q. $\mathrm{If}y=\left(\begin{array}{ccc}\sin x& \cos x& \sin x\\ \cos x& -\sin x& \cos x\\ x& 1& 1\end{array}\right|,\mathrm{then}\frac{\mathrm{dy}}{\mathrm{dx}}=$ .

1. -1
2. cos 2x
3. \N
4. 1

Q. $\frac{dy}{dx}=\tan \left(x+y\right)$

Q. $\int e^{2x}\sin xdx$

Q. $\mathrm{If}x=\cos t\left(3-2{\cos }^{2}t\right),y=\sin t\left(3-2{\sin }^{2}t\right)\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\frac{dy}{dx}\mathrm{at}t=\frac{\mathrm{\pi }}{4}$

Q. $\int {\sin }^3\left(2x+1\right)dx$

Q. $\mathrm{If}y=\left(\begin{array}{ccc}\sin x& \cos x& \sin x\\ \cos x& -\sin x& \cos x\\ x& 1& 1\end{array}\right|,\mathrm{then}\frac{\mathrm{dy}}{\mathrm{dx}}=$ .

1. -1
2. cos 2x
3. \N
4. 1

Q.

Q. $\int \left(e^{\log x}+\sin x\right)\cos xdx$

Q. $\int x^2\cos 2xdx$

Q. Q.32. Differentiate with respect to x
$\frac{1+\sin x}{1-\sin x}$

Q.

find sin ⁶ tita - cos ⁶ tita

Q. $\int \frac{\cos x-\sin x}{1+\sin 2x}dx$

Q. $\int \frac{{\sin }^{3}x}{\sqrt{\cos x}}dx$

Q.

Q. Differentiate with respect to x :
tan^-1 ( 2x / 1 + 15x² )
The answer: 5 /( 1 + 25x² ) - 3 / ( 1 + 9x² )

Q. $\frac{dy}{dx}-y\tan x=e^x$