Higher Order Derivatives

Q. If $y^2=p\left(x\right)$ is a polynomial of degree 3, then $2\frac{d}{dx}$ $\left[y^3\frac{d^{2}y}{d{x}^2}\right]$ is equal to

1. p (x). p"' (x)
2. None of these
3. p"' (x) + p' x
4. p''(x). p"'(x)

Q. $Ify=ta{n}^{-1}\left(\frac{lo{g}_e\left(e/x^2\right)}{lo{g}_e\left(e{x}^2\right)}\right)+ta{n}^{-1}\left(\frac{3+2lo{g}_{e}x}{1-6lo{g}_{e}x}\right),then\frac{d^{2}y}{d{x}^2}is$

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

Q. If $f\left(x\right)=(1+x{)}^n$, then the value of $f\left(0\right)+f^{\prime }\left(0\right)+\frac{f^{\prime \prime }\left(0\right)}{2!}+.....+\frac{f^n\left(0\right)}{n!}$ is

1. $2^{n-1}$
2. $Noneofthese$
3. $n$
4. $2^n$

Q. If $\frac{d^{2}x}{d{y}^2}{\left(\frac{dy}{dx}\right)}^3+\frac{d^{2}y}{d{x}^2}=k$, then k is equal to

1. 0
2. 1
3. 2
4. None of these

Q. If $y=\left(\frac{ax+b}{cx+d}\right)$ , then $2\frac{dy}{dx}.\frac{d^{3}y}{d{x}^3}$ is equal to

1. ${\left(\frac{d^{2}y}{d{x}^2}\right)}^2$
2. $3\frac{d^{2}y}{d{x}^2}$
3. $3{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$
4. $3\frac{d^{2}x}{d{y}^2}$

Q. If $y^2=a{x}^2+bx+c,then{y}^3.\frac{d^{2}y}{d{x}^2}$ is

1. a function of y only
2. a function of x and y
3. a constant
4. a function of x only

Q. $If\sqrt{(x^2+y^2)}=a.e^{ta{n}^{-1\left(y/x\right)}}a>0,theny"\left(0\right)isequalto$

1. 
   $a{e}^{\pi /2}$
2. $-\frac{2}{a}{e}^{-\pi /2}$
3. $\frac{2}{a}{e}^{-\pi /2}$
4. Does not exist

Q. $If{y}^{1/n}=\{x+\sqrt{(1+x^2)}\},$ $then(1+x^2)y_2+x{y}_{1}isequalto$

1. $n^{2}y$
2. $n{y}^2$
3. $n^2{y}^2$
4. $Noneofthese$

Q. If $y=ln{\left(\frac{x}{a+bx}\right)}^x$, then $x^3\frac{d^{2}y}{d{x}^2}$ is equal to

1. ${(\frac{dy}{dx}-y)}^2$
2. ${(\frac{dy}{dx}+x)}^2$
3. ${(x\frac{dy}{dx}-y)}^2$
4. ${(x\frac{dy}{dx}+y)}^2$

Q. The differential equation of the family of the curves $y=e^x(A\cos x+B\sin x)$, where $A$ and $B$ are arbitrary constants is

1. $\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2+y=0$
2. $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y=0$
3. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}-2y=0$
4. $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+y=0$