Chain Rule of Differentiation

Q. If $y=\sin \left(\ln x\right)$, then $\frac{dy}{dx}$ will be

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

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

1. $-\left(2x\right)\sin \left(x^2\right)$
2. $\left(2x\right)\sin \left(x^2\right)$
3. $-\left(x^2\right)\sin \left(x^2\right)$
4. $\left(x^2\right)\sin \left(x^2\right)$

Q.

$\sin (\pi +\theta )\sin (\pi -\theta ){\text{cosec}}^2\theta =$

1. $1$

2. $-1$

3. $\sin \theta$

4. $-\sin \theta$

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.

The value of the integral $\frac{\sin \theta \sin 2\theta ({\sin }^6\theta +{\sin }^4\theta +{\sin }^2\theta )\sqrt{2{\sin }^4\theta +3{\sin }^2\theta +6}}{1-\cos 2\theta }d\theta$ (where $c$ is a constant of integration)

1. $\frac{1}{18}{\left[9–2{\sin }^6\theta –3{\sin }^4\theta –6{\sin }^2\theta \right]}^{\frac{3}{2}}+c$

2. $\frac{1}{18}{\left[11–18{\sin }^2\theta +9{\sin }^4\theta –2{\sin }^6\theta \right]}^{\frac{3}{2}}+c$

3. $\frac{1}{18}{\left[11–18{\cos }^2\theta +9{\cos }^4\theta –2{\cos }^6\theta \right]}^{\frac{3}{2}}+c$

4. $\frac{1}{18}{\left[9–2{\cos }^6\theta –3{\cos }^4\theta –6{\cos }^2\theta \right]}^{\frac{3}{2}}+c$

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. 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 $\frac{dy}{dx}$ of $y=\tan x^2$.

1. $4x{\sec }^2{x}^2$
2. $2x{\tan }^2{x}^2$
3. $2x{\sec }^{2}x$
4. $2x{\sec }^2{x}^2$

Q. If $y={\sin }^2\theta +{\cos }^3\theta$. Find $\frac{dy}{d\theta }$.

1. $2\cos \theta -3{\sin }^2\theta$
2. $2\sin \theta {\cos }^2\theta -3{\cos }^2\theta$
3. $2\sin \theta \cos \theta -3\sin \theta {\cos }^2\theta$
4. $2\sin \theta \cos \theta -3{\sin }^2\theta \cos \theta$

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. Derivative of $f\left(x\right)=\sin (x^3+2x+1)$ is

1. $\cos (3x^2+2)$
2. $\cos (x^3+2x+1)(3x^2+2)$
3. $\cos (x^3+2x+1)(3x^2-2)$
4. $\cos (x^3+2x+1)$

Q.

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

1. $e^{4x}cos2x$

2. $4e^{4x}sin2x$

3. $2e^{4x}[cos2x+2sin2x]$

4. none of these

Q. Differentiation of $\sin x^2$ w.r.t. $x$ is

1. $\cos x^2$
2. $2x\cos x^2$
3. $x^2\cos \left(x^2\right)$
4. $-\cos 2x$

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

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

Q. Find the derivative of $y=(x^2+1)(x^3+3)$.

1. $5x^4+3x^2+6x$
2. $x^4+3x^2+6x$
3. $3x^4+3x^2+x$
4. $2x^4+3x^2+6x$

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

Q. Find the derivative of $y=(x^2+1)(x^3+3)$.

1. $x^4+3x^2+6x$
2. $5x^4+3x^2+6x$
3. $3x^4+3x^2+x$
4. $2x^4+3x^2+6x$

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

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.

If $x=a{t}^4,y=b{t}^3,then\frac{dy}{dx}=?$

1. $\frac{3b}{4at}$

2. $\frac{4at}{3b}$

3. 12abt

4. none of these

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

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

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

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

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

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

Q.

Prove the given identity by using the identity $1+{\cos }^2\theta =\cos e{c}^2\theta$:

${\left(\mathrm{sinA}+\mathrm{secA}\right)}^2+{(\mathrm{cosA}+\mathrm{cosecA})}^2={(1+\mathrm{secA}\mathrm{cosecA})}^2$.

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

Q.

$y=cos{x}^3,then\frac{dy}{dx}$=?

1. $-3xsin{x}^3$

2. $sin{x}^3$

3. $\frac{sin{x}^3}{x^2}$

4. none of these

Q. If $y={\sin }^2\theta +{\cos }^3\theta$. Find $\frac{dy}{d\theta }$.

1. $2\cos \theta -3{\sin }^2\theta$
2. $2\sin \theta {\cos }^2\theta -3{\cos }^2\theta$
3. $2\sin \theta \cos \theta -3\sin \theta {\cos }^2\theta$
4. $2\sin \theta \cos \theta -3{\sin }^2\theta \cos \theta$