Derivative of Standard Inverse Trigonometric Functions

Q. If y = $ta{n}^{-1}\sqrt{\left(\frac{1+sinx}{1-sinx}\right)},\frac{\pi }{2}<x<\pi$, then $\frac{dy}{dx}$ equals

1. $\frac{-1}{2}$
2. $-1$
3. $\frac{1}{2}$
4. $1$

Q. Match the following functions to their derivatives?
$\begin{array}{cc}Function& Derivatives\\ a)si{n}^{-1}x& 1)\frac{-1}{|x|\sqrt{x^2-1}}\\ b)co{s}^{-1}x& 2)\frac{-1}{1+x^2}\\ c)ta{n}^{-1}x& 3)\frac{1}{|x|\sqrt{x^2-1}}\\ d)se{c}^{-1}x& 4)\frac{1}{1+x^2}\\ e)co{t}^{-1}x& 5)\frac{-1}{\sqrt{1-x^2}}\\ f)cose{c}^{-1}x& 6)\frac{1}{\sqrt{1-x^2}}\end{array}$

1. a-6, b-5, c-4, d-3, e-2, f-1
2. a-5, b-6, c-4, d-3, e-2, f-1
3. a-6, b-5, c-3, d-4, e-2, f-1
4. a-6, b-5, c-3, d-2, e-1, f-2

Q. Match the following functions to their derivatives?
$\begin{array}{cc}Function& Derivatives\\ a)si{n}^{-1}x& 1)\frac{-1}{|x|\sqrt{x^2-1}}\\ b)co{s}^{-1}x& 2)\frac{-1}{1+x^2}\\ c)ta{n}^{-1}x& 3)\frac{1}{|x|\sqrt{x^2-1}}\\ d)se{c}^{-1}x& 4)\frac{1}{1+x^2}\\ e)co{t}^{-1}x& 5)\frac{-1}{\sqrt{1-x^2}}\\ f)cose{c}^{-1}x& 6)\frac{1}{\sqrt{1-x^2}}\end{array}$

1. a-6, b-5, c-4, d-3, e-2, f-1
2. a-5, b-6, c-4, d-3, e-2, f-1
3. a-6, b-5, c-3, d-4, e-2, f-1
4. a-6, b-5, c-3, d-2, e-1, f-2

Q. If $\sqrt{3}\cos x+\sin x=\sqrt{2}$, then general value of ​x is
(a) $n\pi +{\left(-1\right)}^n\frac{\pi }{4},n\in Z$

(b) ${\left(-1\right)}^n\frac{\pi }{4}-\frac{\pi }{3},n\in Z$

(c) $n\pi +\frac{\pi }{4}-\frac{\pi }{3},n\in Z$

(d) $n\pi +{\left(-1\right)}^n\frac{\pi }{4}-\frac{\pi }{3},n\in Z$

Q. $\mathrm{If}y={\sin }^{-1}\left(\cos x\right),\mathrm{where}x\in \left(0,2\pi \right),\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\mathrm{dy}}{\mathrm{dx}}\mathrm{is}$

1. $-1,if0<x<\pi$
2. $1,if\pi <x<2\pi$
3. $1,if0<x<\pi$
4. $-1,if\pi <x<2\pi$

Q. Prove that:
$\cos 4x-\cos 4\alpha =8\left(\cos x-\cos \alpha \right)\left(\cos x+\cos \alpha \right)\left(\cos x-\sin \alpha \right)\left(\cos x+\sin \alpha \right)$

Q. Write the values of x in [0, π] for which $\sin 2x,\frac{1}{2}$ and cos 2x are in A.P.

Q.

If y = $sec\left(ta{n}^{-1}x\right)$, then $\frac{dy}{dx}$ at x = 1 is equal to:

1. $\frac{1}{\sqrt{2}}$

2. $\frac{1}{2}$

3. $1$

4. $0$

Q. If $f\left(x\right)=co{t}^{-1}\left(\frac{x^x-x^{-x}}{2}\right)$, then $f^{\prime }\left(1\right)$ equals

1. -1
2. 1
3. log 2
4. -log 2

Q. Prove that:
(i) $\cos \left(\frac{3\mathrm{\pi }}{4}+x\right)-\cos \left(\frac{3\mathrm{\pi }}{4}-x\right)=-\sqrt{2}\sin x$

(ii) $\cos \left(\frac{\mathrm{\pi }}{4}+x\right)+\cos \left(\frac{\mathrm{\pi }}{4}-x\right)=\sqrt{2}\cos x$

Q.

If y = $sec\left(ta{n}^{-1}x\right)$, then $\frac{dy}{dx}$ at x = 1 is equal to:

1. $\frac{1}{\sqrt{2}}$

2. $\frac{1}{2}$

3. $1$

4. $0$

Q. Write the value of $\underset{x\to \mathrm{\pi }/2}{\lim }\frac{2x-\mathrm{\pi }}{\cos x}.$