Differentiation Using Substitution

Q. If $y=si{n}^{-1}\frac{\sqrt{(1+x)}+\sqrt{(1-x)}}{2}$, then $\frac{dy}{dx}=$

1. $\frac{1}{\sqrt{(1-x^2)}}$
2. $-\frac{1}{\sqrt{(1-x^2)}}$
3. $-\frac{1}{2\sqrt{(1-x^2)}}$
4. None of these

Q. Let $y=\left({\cot }^{-1}x\right)\left({\cot }^{-1}\right(-x\left)\right)$ and range of $y\in (0,\frac{a{\pi }^2}{b}]$, then the value of $a+b$ is

1. $4$
2. $2$
3. $6$
4. $5$

Q. If $co{s}^{-1}\left(\frac{x^2-y^2}{x^2+y^2}\right)=loga$ then $\frac{dy}{dx}$ is equal to

1. $\frac{y}{x}$
2. $\frac{x}{y}$
3. $\frac{x^2}{y^2}$
4. $\frac{y^2}{x^2}$

Q. If $\sqrt{(1-x^{2n})}+\sqrt{(1-y^{2n})}=a(x^n-y^n)$, then $\sqrt{\left(\frac{1-x^{2n}}{1-y^{2n}}\right)}\frac{dy}{dx}$ is equal to

1. $\frac{x^{n-1}}{y^{n-1}}$
2. $\frac{y^{n-1}}{x^{n-1}}$
3. $\frac{x}{y}$
4. $1$

Q. The derivative of f(x) defined by $f\left(x\right)={\tan }^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right),-\pi <x<\pi$
is

1. $1,\mathrm{if}0<x<\pi$
2. $-1,\mathrm{if}-\pi <x<0$
3. $\frac{1}{2},\mathrm{if}0<x<\pi$
4. $-\frac{1}{2},\mathrm{if}-\pi <x<0$

Q. $Ify=\sqrt{\left(\frac{1+cos2\theta }{1-cos2\theta }\right)},\frac{dy}{d\theta }at\theta =\frac{3\pi }{4}is$

1. $-2$
2. $2$
3. $\pm 2$
4. None of these

Q. The differentiation of ${\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)w.r.t.{\tan }^{-1}x$ is

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

Q.

If $\sqrt{1-x^{2n}}+\sqrt{1-y^{2n}}=a(x^n-y^n)$ then $\sqrt{\frac{1-x^{2n}}{1-y^{2n}}}.\frac{dy}{dx}=$

1. $1$

2. $\frac{x}{y}$

3. $\frac{x^{n-1}}{y^{n-1}}$

4. $\frac{y}{x}$

Q. If $y=ta{n}^{-1}\frac{\sqrt{1+x^2}-1}{x}$, then

1. $y^{\prime }\left(1\right)=1$
2. $y^{\prime }\left(1\right)=\frac{1}{4}$
3. $y^{\prime }\left(1\right)=0$
4. $y^{\prime }\left(1\right)$ does not exist.