Chain Rule of Differentiation

Q.

The value of$co{t}^{-1}\left(9\right)+\cos e{c}^{-1}\left(\frac{\sqrt{41}}{4}\right)$ is given by

1. $0$

2. ${\tan }^{-1}x$

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

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

Q.

$\int \frac{x^2-1}{(x^4+3x^2+1){\tan }^{-1}\left(x+{\displaystyle \frac{1}{x}}\right)}dx=$

1. ${\tan }^{-1}\left(x+\frac{1}{x}\right)+c$

2. $co{t}^{-1}\left(x+\frac{1}{x}\right)+c$

3. $\log \left(x+\frac{1}{x}\right)+c$

4. $\log \left({\tan }^{-1}\left(x+\frac{1}{x}\right)\right)+c$

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.

Two AMs $A_1$ and $A_2$, two GMs $G_1$ and $G_2$ and two HMs $H_1$ and $H_2$ are inserted between two numbers a and b, then

$\frac{1}{H_1}$ + $\frac{1}{H_2}$ equals.

1. 

2. 

3. 

4. 

Q. Find the derivative of the following function:
f(x) = $cosecxcotx$

Q. 히ㅎThe derivative of function f(x) = log (2x) w.r.t. tis1 dxхрP LY f(X) 2x dt= (x), хр.(4) f dt1 dx(3) f(X)=d:

Q.

Let $f\left(x\right)={\tan }^{-1}x,$ then $f\left(x\right)+f\left(x\right)$ is equal to $0$, where $x$is equal to

1. $0$

2. $1$

3. $i$

4. $-i$

Q.

The derivative of at x = 100 is:

1. 100

2. 0

3. 99

4. 9900

Q. Prove that:
(i) $\sin \left(\frac{\pi }{3}-x\right)\cos \left(\frac{\pi }{6}+x\right)+\cos \left(\frac{\pi }{3}-x\right)\sin \left(\frac{\pi }{6}+x\right)=1$
(ii) $\sin \left(\frac{4\mathrm{\pi }}{9}+7\right)\cos \left(\frac{\mathrm{\pi }}{9}+7\right)-\cos \left(\frac{4\mathrm{\pi }}{9}+7\right)\sin \left(\frac{\mathrm{\pi }}{9}+7\right)=\frac{\sqrt{3}}{2}$
(iii) $\sin \left(\frac{3\mathrm{\pi }}{8}-5\right)\cos \left(\frac{\mathrm{\pi }}{8}+5\right)+\cos \left(\frac{3\mathrm{\pi }}{8}-5\right)\sin \left(\frac{\mathrm{\pi }}{8}+5\right)=1$

Q. If $z=\frac{1}{1-\mathrm{cos\theta }-i\mathrm{sin\theta }}$, then Re (z) =
(a) 0
(b) $\frac{1}{2}$
(c) $\cot \frac{\mathrm{\theta }}{2}$
(d) $\frac{1}{2}\cot \frac{\mathrm{\theta }}{2}$

Q. Let $f:R\to R$ be a function defined by $f\left(x\right)=x^3+x^2+x-1.$ If $g$ is the inverse of $f,$ then $g^{\prime }\left(2\right)$ is

1. $2$
2. $6$
3. $\frac{1}{6}$
4. $\frac{1}{2}$

Q. If $f_{n-1}\left(x\right)=\ln \left(f_n\right(x\left)\right)\forall n\in N$ and $f_0\left(x\right)=x-1$, then $\frac{d}{dx}(f_n\left(x\right))$ is

1. $f_n\left(x\right)\cdot f_{n-1}\left(x\right)$
2. $f_n\left(x\right)\cdot f_{n-1}\left(x\right)\cdot f_{n-2}\left(x\right)\cdots f_1\left(x\right)$
3. $f_{n-1}\left(x\right)\cdot \frac{d}{dx}(f_{n-1}\left(x\right))$
4. $f_n\left(x\right)\cdot \frac{d}{dx}(f_{n-1}\left(x\right))$

Q. derivative of x² sin 2x

Q. If $f$ is an invertible function such that $f\left(x\right)=x^3+e^{x/2}$ and $g\left(x\right)=f^{-1}\left(x\right)\forall x,$ then the value of $g^{\prime }\left(1\right)$ is

Q. find the derivative of y=sin3x

Q.

Find $\frac{d}{dx}\left({\cos }^{2}x\right)$ by using chain rule.

Q. If $y=f\left(x\right)\text{satisfies}2e^{x{y}^2}+ycos\left(x^2\right)=4$ then value of $\left|\frac{f^{\prime }\left(0\right)}{5}\right|$ is

Q. Let $f:R\to R,g:R\to R$ and $h:R\to R$ be differentiable functions such that $f\left(x\right)=x^3+3x+2,g\left(f\right(x\left)\right)=x$ and $h\left(g\right(g\left(x\right)\left)\right)=x$ for all $x\in R.$ 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. Let $y=f\left(x\right)$ is a positive function which satisfies equation $\sqrt{y^2+2x}+\sqrt{y^2-2x}=2x^2,\text{then}\frac{dy}{dx}$ is

1. $\frac{2x^3-x^{-3}}{y}$
2. $\frac{2x^4-x^{-2}}{y}$
3. $\frac{2x^4-x^{-2}}{\sqrt{x^6+1}}$
4. $\frac{2x^3-x^{-3}}{\sqrt{1+x^6}}$

Q. Find the derivative of the following function (it is to be understood that a, is a fixed non-zero constant):
f(x)= (x+a)

Q. If $f\left(x\right)=\sqrt{x^2+6x+9}$, then f '(x) is equal to

1. $1forx>-3$
2. $-1forx<-3$
3. $1forallx\in R$
4. $0forx=-3$

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. The minimum value of $x^2+\frac{1}{1+x^2}$ occurs at

1. $x=4$
2. $x=3$
3. $x=0$
4. $x=1$

Q.

What is the Derivative of : ( ax + b )ⁿ ?

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. Find its derivative by using 1st principle--e^sinx

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

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

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

Q. Suppose $f$ and $g$ are differentiable functions on $(0,\infty )$ such that $f\text{'}\left(x\right)=-\frac{g\left(x\right)}{x}$ and $g\text{'}\left(x\right)=-\frac{f\left(x\right)}{x}$, for all $x>0$. Further, $f\left(1\right)=3$ and $g\left(1\right)=-1$.

$f\left(10\right)$ is equal to

1. $\frac{101}{20}$
2. $10$
3. $20$
4. $\frac{201}{20}$

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. $1+x^5$

4. $5x^4$

Q. The value(s) of $x$ for which the value of $detA=0,$ where $A=\left[\begin{array}{ccc}x-1& 1& 1\\ 1& x-1& 1\\ 1& 1& x-1\end{array}\right]$

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