Chain Rule of Differentiation

Q. $Iff\left(x\right)=\left(lo{g}_{cotx}tanx\right)(lo{g}_{tanx}cotx{)}^{-1}+ta{n}^{-1}\left(\frac{x}{\sqrt{(4-x^2)}}\right)$ then f'(0) is equal to

1. $-2$
2. $0$
3. $\frac{1}{2}$
4. $2$

Q. If $f\left(x\right)=x|x|$, then the value of $f^{\prime }\left(x\right)-2|x|$ at $x=2021$ is

Q. Derivative of $y=(sinx{)}^2$ with respect to x will be

1. 
2. 
3. 
4. 

Q. The derivative of $y=lo{g}_{e}sin\left(e^x\right)$ with respect to x will be

1. 
2. 
3. 
4. 

Q. $Ify={\tan }^{-1}\left[\frac{2^x}{1+2^{x+1}}\right],-\infty <x<\infty then\frac{dy}{dx}atx=0is$

1. $\frac{-3}{5}\ln 2$
2. $\frac{1}{10}\ln 2$
3. $2$
4. None of these

Q.

If $g$ is the inverse of a function $f$ and $f^{\prime }\left(x\right)=\frac{1}{1+x^5}$, then $g^{\prime }\left(x\right)$ is equal to:

1. $\frac{1}{1+\left\{g\right(x){\}}^5}$

2. ${1+\left\{g\right(x\left)\right\}}^5$

3. $5x^4$

4. $1+x^5$

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.

Let $f:\to R\to R,g:R\to Randh:R\to R$ be differentiable functions such that $f\left(x\right)=x^3+3x+2,g\left(f\right(x\left)\right)=xandh\left(g\right(g\left(x\right)\left)\right)=xforallx\epsilon R.$ Then

1. $g^{\prime }\left(2\right)=\frac{1}{15}$

2. $h^{\prime }\left(1\right)=666$

3. $h\left(0\right)=16$

4. $h\left(g\right(3\left)\right)=36$

Q.

If $F\left(x\right)={\left(f\left(\frac{x}{2}\right)\right)}^2+{\left(g\left(\frac{x}{2}\right)\right)}^2$ where $f^{\prime }\left(x\right)=-f\left(x\right)$ and $g\left(x\right)=f^{\prime }\left(x\right)$ and given that $F\left(5\right)=5$, then $F\left(10\right)$ is equal to

1. 5

2. 10

3. 0

4. 15

Q.

If $2x=y^{1/5}+y^{-1/5}$ then $(x^2-1)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=$

1. $25y^2$

2. y + 25

3. 5y

4. 25y