Chain Rule of Differentiation

Q.

What is the derivative of ${\sin }^2\left(x\right)$?

Q.

Nine days ago the area covered by the piece of bread was $3inc{h}^2$. Total, the mould covers $9inc{h}^2$. What is the rate of change in moulds area?

Q.

Prove that $\sin 20\times \sin 40\times \sin 60\times \sin 80=\frac{3}{16}$

Q.

What is the derivative of ${\sin }^{-1}\left(x\right)$?

Q. If surface area of a cube is changing at a rate of $5m^2/s$ , find the rate at which length of the body diagonal changes, at a moment when side length is $1m$.

1. $5\text{m/s}$
2. $\frac{5}{4\sqrt{3}}\text{m/s}$
3. $\frac{5}{2}\text{m/s}$
4. $5\sqrt{3}\text{m/s}$

Q. If $S=e^x\sin x$, find $\frac{d^{2}S}{d{x}^2}$

1. $2e^x\sin x$
2. $\frac{e^x\cos x}{2}$
3. $2e^x\cos x$
4. $\frac{e^x\sin x}{2}$

Q. Derivative of $f\left(x\right)=x\sin \left(\sqrt{x}\right)$ is

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

Q. Find derivative of $f\left(x\right)=e^{\sin x}+\ln x+8^x$

1. $\cos x{e}^{\sin x}+\frac{1}{x}+8^x\ln x$
2. $\cos x{e}^{\sin x}+\frac{1}{x}+8^x\ln 8$
3. $\sin x{e}^{\sin x}+\frac{1}{x}+8^x\ln 8$
4. $\cos x{e}^{\cos x}+\frac{1}{x}+8^x\ln x$

Q.

Two boys raising a load pull at an angle to each other. If they exert forces of 30 N and 60 N respectively and their effective pull is at right angles to the directions of the pull of the first boy, what is the angle between their arms? What is the effective pull.

Q. If $y={\sin }^5\theta +{\cos }^8\theta$. Find $\frac{dy}{d\theta }$.

1. ${\sin }^4\theta {\cos }^3\theta +\tan \theta$
2. ${\sin }^4\theta {\cos }^3\theta +{\tan }^5\theta$
3. ${\sin }^2\theta {\cos }^3\theta +{\tan }^5\theta$
4. $5\cos \theta {\sin }^4\theta -8\sin \theta {\cos }^7\theta$

Q.

The domain of the derivative of the function $f\left(x\right)={\tan }^{-1}x$. if $\left|x\right|\le 1=\left(\frac{1}{2}\right)\left[\left|x\right|-1\right]$ if $\left|x\right|>1$ is

1. $R-\left\{0\right\}$

2. $R-\left\{1\right\}$

3. $R-\left\{-1\right\}$

4. $R-\left\{-1,1\right\}$

Q.

The first derivative of the function $f\left(x\right)=\sin \left(2x\right)\cos \left(2x\right)\cos \left(3x\right)+{\log }_2{2}^{x+3}$ with respect to $x$ at $x=\pi$ is

1. $2$

2. $-1$

3. $1$

4. None of these

Q. Find the derivative of $y=e^{\sin \theta }$ with respect to $\theta$

1. $e^{\sin \theta }\left(\cos \theta \right)$
2. $e^{\cos \theta }\left(\sin \theta \right)$
3. $e^{\cos \theta }\left(\cot \theta \right)$
4. $e^{\tan \theta }\left(\sin \theta \right)$

Q. Derivative of $f\left(x\right)=\frac{e^x}{\cos \left(x^2\right)}$ is

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

Q.

${\cos }^2\left(\alpha \right)+{\cos }^2\left(\alpha +120°\right)+{\cos }^2\left(\alpha -120°\right)$ is equal to

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

2. $1$

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

4. $0$

Q. Find derivative of $f\left(x\right)=e^{\ln x}+\sin x+8$

1. $e^{\ln x}.\frac{1}{x}-\sin x$
2. $e^{\ln x}.\frac{1}{x}-\cos x$
3. $e^{\ln x}.\frac{1}{x^2}+\cos x$
4. $e^{\ln x}.\frac{1}{x}+\cos x$

Q. Derivative of $f\left(x\right)=e^x\sin \left(\sqrt{x}\right)$ is

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

Q. Find derivative of $f\left(x\right)=e^{\ln x}+\sin x+8$

1. $e^{\ln x}.\frac{1}{x^2}+\cos x$
2. $e^{\ln x}.\frac{1}{x}-\sin x$
3. $e^{\ln x}.\frac{1}{x}+\cos x$
4. $e^{\ln x}.\frac{1}{x}-\cos x$

Q. Find the derivative of function $y=x\cos x$.

1. $\cos x-x\sin x$
2. $\cos x-\sin x$
3. $x\cos x-\sin x$
4. None of the above

Q.

If $y=e^{4x}sin2x$, then $\frac{dy}{dx}$=?

1. 

2. none of these

3. 

4. 

Q. If $y=\tan \left[\text{log}\right(x^2\left)\right]$, then $\frac{dy}{dx}$ is -

1. $2x{\sec }^2\left[\text{log}\right(x^2\left)\right]$
2. $\frac{2{\sec }^2\left[\text{log}\right(x^2\left)\right]}{x}$
3. $\frac{1}{x^2}{\sec }^2\left[\text{log}\right(x^2\left)\right]$
4. $x{\sec }^2\left[\text{log}\right(x^2\left)\right]$

Q.

What is the derivative of ${\sin }^{-1}\left(2x\sqrt{1-x^2}\right)$ with respect to ${\sin }^{-1}\left(3x-4x^3\right)$?

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

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

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

4. $1$

Q.

${\int }_0^{2\pi }\left(\sin \left(x\right)+\left|\sin \left(x\right)\right|\right)dx=$

1. $0$

2. $4$

3. $8$

4. $1$

Q. If $y=\cos x^3$, then find $\frac{dy}{dx}$ at $x=0$

1. $0$
2. $1$
3. $-3$
4. $3$

Q. If $y=\cos x^3$, then find $\frac{dy}{dx}$ at $x=0$

1. $0$
2. $1$
3. $-3$
4. $3$

Q.

If $y=\frac{{\sin }^{-1}x}{\sqrt{1-x^2}}$, then $(1-x^2)\frac{dy}{dx}$is equal to

1. $x+y$

2. $1+xy$

3. $1-xy$

4. $xy-2$

Q. If $f\left(x\right)=x\cos x$, find $f^{\prime \prime }\left(x\right)$.

1. $-\cos x-2\sin x$
2. $-2x\sin x-2\cos x$
3. $-x\sin x-2\cos x$
4. $-x\cos x-2\sin x$

Q. Find the derivative of $y$ with respect to $x$ at $x=1$, where function $y$ is expressed as $y=\sqrt{x^3+1}$ .

1. $\frac{3}{\sqrt{2}}$
2. $\frac{3}{2\sqrt{2}}$
   
3. $\frac{3\sqrt{2}}{\sqrt{5}}$
4. $\frac{1}{2\sqrt{2}}$
   

Q.

If $\left|x\right|<1$, $\left|y\right|<1$, then the sum of infinity of the following series: $\left(x+y\right)+\left(x^2+xy+y^2\right)+\left(x^3+x^{2}y+x{y}^2+y^3\right)+......$ is

1. $\frac{x+y+xy}{\left(1-x\right)\left(1-y\right)}$

2. $\frac{x+y-xy}{\left(1-x\right)\left(1-y\right)}$

3. $\frac{x+y+xy}{\left(1+x\right)\left(1+y\right)}$

4. $\frac{x+y-xy}{\left(1+x\right)\left(1+y\right)}$

Q. Find the derivative of $y=e^{\sin \theta }$ with respect to $\theta$

1. $e^{\cos \theta }\left(\sin \theta \right)$
2. $e^{\sin \theta }\left(\cos \theta \right)$
3. $e^{\tan \theta }\left(\sin \theta \right)$
4. $e^{\cos \theta }\left(\cot \theta \right)$