Chain Rule of Differentiation

Q.

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

Q. If $y=\sin x$ and $x=3t$, then $\frac{dy}{dt}$ will be

1. $9\cos x$
2. $\cos x$
3. $3\cos \left(3t\right)$
4. $-\cos x$

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

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

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.

Multiply $\left(x+5\right)\left(x-3\right)$.

Q. Find the differentiation of a function represented by $y=e^{x^3}$ with respect to $x$.

1. $e^{x^3}.3x^3$
2. $e^{x^2}.3x^2$
3. $e^{x^3}\times 3x^2$
4. $e^{x^3}.3x$

Q.

If $y=\frac{\sin (x+a)}{\sin (x+b)}$, $a\ne b$, then $y$ is

1. minimum at $x=0$

2. maxima at $x=0$

3. neither minima nor maxima at $x=0$

4. None of the above

Q.

If $a=\cos \theta +i\sin \theta$, then $\frac{(1+a)}{(1-a)}$ is equal to

1. $cot\theta$

2. $cot\left(\frac{\theta }{2}\right)$

3. $icot\left(\frac{\theta }{2}\right)$

4. $i\tan \left(\frac{\theta }{2}\right)$

Q.

Find the derivative of $y=\sqrt{x^2+1}$

1. 

2. 

3. 

4. none of these

Q.

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

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.

Find the derivative of $y=\sqrt{x^2+1}$

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

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

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

4. none of these

Q.

Multiply $\left(x+3\right)\left(x-7\right)$.

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.

Add the following:

$7m,n,2m,3n$

Q. Find the differentiation of a function represented by $y=e^{x^3}$ with respect to $x$.

1. $e^{x^3}.3x^3$
2. $e^{x^2}.3x^2$
3. $e^{x^3}\times 3x^2$
4. $e^{x^3}.3x$

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

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

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.

How do you combine like terms for $3x^2+21x+5x+10x^2$?

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.

Add the following:

$4x^{2}y,-3x{y}^2,-5x{y}^2,2x^{2}y$