Chain Rule of Differentiation

Q.

If $f\left(x\right)=2x$ and $g$ is identity function, then:

1. $\left(fog\right)\left(x\right)=g\left(x\right)$

2. $\left(gog\right)\left(x\right)=g\left(x\right)$

3. $\left(fog\right)\left(x\right)=(g+g)\left(x\right)$

4. $\left(fog\right)\left(x\right)=(f+f)\left(x\right)$

Q.

Find the value of $\sin {270}^{\circ }$.

Q.

How do you find the derivative of $y=\cos \left(2x\right)$ ?

Q. how to find the integral of sin inverse of x

Q.

Calculate the value of $\sin \left(22.5\right)$. Show the steps.

Q.

${\sin }^2\left(\frac{\pi }{8}\right)+{\sin }^2\left(\frac{3\pi }{8}\right)+{\sin }^2\left(\frac{5\pi }{8}\right)+{\sin }^2\left(\frac{7\pi }{8}\right)=$

1. $1$

2. $-1$

3. $0$

4. $2$

Q.

The value of ${\cos }^{-1}\left(\cos \left(\frac{5\pi }{3}\right)\right)+{\sin }^{-1}\left(\sin \left(\frac{5\pi }{3}\right)\right)$

1. $0$

2. $\frac{\pi }{2}$

3. $\frac{2\pi }{3}$

4. $\frac{10\pi }{3}$

Q.

What is the formula of $\cos 3x$?

Q.

$\frac{d}{dx}\left(x^2\sin \left(\frac{1}{x}\right)\right)=$

1. $\cos \left(\frac{1}{x}\right)+2x\sin \left(\frac{1}{x}\right)$

2. $2x\sin \left(\frac{1}{x}\right)-\cos \left(\frac{1}{x}\right)$

3. $\cos \left(\frac{1}{x}\right)-2x\sin \left(\frac{1}{x}\right)$

4. none of these

Q.

What is the integration of ${\sin }^{4}x$?

Q.

Find the value of $\sin {240}^{\circ }$.

Q.

The relation $\mathrm{R}$ defined on the set $\mathrm{A}=\left\{1,2,3,4,5\right\}$ by $\mathrm{R}=\left\{(\mathrm{x},\mathrm{y}):|{\mathrm{x}}^2–{\mathrm{y}}^2|<16\right\}$ is given by

1. $\left\{(1,1),(2,1),(3,1),(4,1),(2,3)\right\}$

2. $\left\{(2,2),(3,2),(4,2),(2,4)\right\}$

3. $\left\{(3,3),(3,4),(5,4),(4,3),(3,1)\right\}$

4. None of these

Q.

Find the derivative of the given function: $\sin (3x+5)$.

Q.

$\frac{d}{dx}\sqrt{\left(\frac{1-\sin 2x}{1+\sin 2x}\right)}$is equal to

1. $se{c}^{2}x$

2. $-se{c}^2\left(\frac{\pi }{4}–x\right)$

3. $se{c}^2\left(\frac{\pi }{4}+x\right)$

4. $se{c}^2\left(\frac{\pi }{4}–x\right)$

Q.

The differential coefficient of $f(\log x)$ when $f\left(x\right)=\log x$ is:

1. $x\log x$

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

3. $\frac{1}{x\log x}$

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

Q.

If $x+\frac{1}{x}=3$ then find $x^5+\frac{1}{x^5}$.

Q.

Factorise: $a^3-\frac{1}{a^3}-2a+\frac{2}{a}$.

Q.

Given that the inverse trigonometric functions take principal values only.

Then, the number of real values of $x$ which satisfy ${\sin }^{-1}\left(\frac{3x}{5}\right)+{\sin }^{-1}\left(\frac{4x}{5}\right)={\sin }^{-1}x$

Q.

Find the value of $\sqrt{2}\sqrt{-3}$:

1. $\sqrt{6}$

2. $-\sqrt{6}$

3. $\mathrm{i}\sqrt{6}$

4. None of these

Q.

If $f\left(x\right)=\frac{2x-1}{x+5}$, $x\ne -5$, then $f^{-1}\left(x\right)=$

1. $\frac{x+5}{2x-1},x\ne \frac{1}{2}$

2. $\frac{5x+1}{2-x},x\ne 2$

3. $\frac{x-5}{2x-1},x\ne \frac{1}{2}$

4. $\frac{5x-1}{2-x},x\ne 2$

Q.

How do you simplify ${\cos }^{4}x-{\sin }^{4}x$?

Q.

Prove that: $ta{n}^{-1}x-ta{n}^{-1}y=ta{n}^{-1}\left(\frac{x-y}{1+xy}\right)$

Q.

Find the product: $\left(5-2x\right)\left(3+x\right)$

Q. Find the derivative of $\frac{4x+5\sin x}{3x+7\cos x}$

Q.

What is the formula of $\sin 4A$

Q. If $2x=y^{1/5}+y^{-1/5}$ and $(x^2-1)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=ky,$ then the value of $k$ is

Q.

Differential coefficient of $\sqrt{\sec \left(\sqrt{x}\right)}$ is

1. $\left(\frac{1}{4\sqrt{x}}\right)\left(\sec \left(\sqrt{x}\right)\right)\left(\sin \left(\sqrt{x}\right)\right)$

2. $\left(\frac{1}{4\sqrt{x}}\right){\left(\sec \left(\sqrt{x}\right)\right)}^{\frac{3}{2}}\left(\sin \left(\sqrt{x}\right)\right)$

3. $\left(\frac{1}{2}\right)\sqrt{x}\left(\sec \left(\sqrt{x}\right)\right)\sin \left(\sqrt{x}\right)$

4. $\left(\frac{1}{2}\right)\sqrt{x}{\left(\sec \left(\sqrt{x}\right)\right)}^{\frac{3}{2}}\sin \left(\sqrt{x}\right)$

Q.

If $\alpha =3{\sin }^{-1}\left(\frac{6}{11}\right)and\beta =3{\cos }^{-1}\left(\frac{4}{9}\right)$, where the inverse trigonometric functions take only the principal values, then the correct option(s) is (are)

1. $\cos \beta >0$

2. $\sin \beta <0$

3. $\cos (\alpha +\beta )>0$

4. $\cos \alpha <0$

Q.

If $y={[x+\sqrt{x^2+a^2}]}^n,then\frac{dy}{dx}$ is equal to

1. $\frac{ny}{\sqrt{x^2+a^2}}$

2. $\frac{nx}{\sqrt{x^2+a^2}}$

3. $-\frac{nx}{\sqrt{x^2+a^2}}$

4. $-\frac{ny}{\sqrt{x^2+a^2}}$

Q.

What is the derivative of an inverse function?