Derivatives

Q.

What is the derivative of $xy$?

Q. Differentiate the following equation,
$lo{g}_{x^2}x$

Q.

If $f\left(x\right)=\left|x-2\right|+\left|x+1\right|-x$, then $f\left(-10\right)$ is equal to

1. $-3$

2. -$-2$

3. $-1$

4. $0$

Q. If $y=\frac{x}{\log |cx|}$ (where $c$ is an arbitrary constant) is the general solution of the differential equation
$\frac{dy}{dx}=\frac{y}{x}+\varphi \left(\frac{x}{y}\right)$, then the function $\varphi \left(\frac{x}{y}\right)$ is

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

Q. Find the values of $x$ for which $y=\left[x\right(x-2){]}^2$ is an increasing function.

Q. Find the differential equation of the family of curves whose equations are $\frac{x^2}{a^2}+\frac{y^2}{a^2+\lambda }=1,$ where $\lambda$ is parameter.

1. $-\frac{xy}{a^2{y}^{\prime }}=\frac{a^2-x^2}{a^2}$
2. $-\frac{xy}{a^2{y}^{\prime }}=\frac{a^2+x^2}{a^2}$
3. $-\frac{xy}{a^2{y}^{\prime }}=\frac{a^4-x^2}{a^2}$
4. $-\frac{xy}{a^2{y}^{\prime }}=\frac{a^4+x^2}{a^2}$

Q. If for the differential equation $y^1=\frac{y}{x}+\varphi \left(\frac{x}{y}\right)$ the general solution is $y=\frac{x}{log|Cx|}$ then $\varphi \left(\frac{x}{y}\right)$is given by

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

Q.

Differentiate the following questions w.r.t. x.

log(log x), x>1.

Q.

If x < 2, then write the value of $\frac{d}{dx}\left(\sqrt{x^2-4x+4}\right)$.

Q. Let $y=y\left(x\right)$ be the solution of the differential equation, $\frac{2+\text{sin}x}{y+1}.\frac{dy}{dx}=-\text{cos}x,y>0,y\left(0\right)=1.$
If $y\left(\pi \right)=a,$ and $\frac{dy}{dx}$ at $x=\pi$ is $b$, then the ordered pair $(a,b)$ is equal to:

1. $(2,\frac{3}{2})$
2. $(1,1)$
3. $(2,1)$
4. $(1,-1)$

Q.

If f(x) = |x| + |x - 1|, write the value of $\frac{d}{dx}\left(f\right(x\left)\right)$.

Q. If $f\left(x\right),g\left(x\right)$ be twice differential functions on $[0,2]$ satisfying $f^{\prime \prime }\left(x\right)=g^{\prime \prime }\left(x\right)$, $f^{\prime }\left(1\right)=2g^{\prime }\left(1\right)=4$ and $f\left(2\right)=3g\left(2\right)=9$, then

1. $f\left(4\right)-g\left(4\right)=10$
   
2. $|f\left(x\right)-g\left(x\right)|<2\Rightarrow -2<x<0$
   
3. $f\left(2\right)=g\left(2\right)\Rightarrow x=-1$
   
4. $f\left(x\right)-g\left(x\right)=2x$ has real root

Q.

Write the value of $\frac{d}{dx}\left\{\right(x+|x|\left)|x|\right\}$

Q. Differentiate the given function w.r.t. $x$.
$\log \left(\log x\right),x>1$

Q.

Write the value of the derivative of $f\left(x\right)=|x-1|+|x-3|\text{at}x=2.$

Q. How do you differentiate $f\left(x\right)=\log \left(\log x\right)$ ?

Q.

Write the value of $\frac{d}{dx}\left(log|x|\right).$

Q.

If $f\left(x\right)=\frac{x^2}{|x|}$, write $\frac{d}{dx}\left(f\right(x\left)\right).$

Q. Find the sum of the fourth powers of the roots of
$x^3-2x^2+x-1=0$.

Q. Find the order and degree of the given differential equation: $y^{\prime }=siny$. The order of this equation is the same as its degree. If true enter 1 else enter 0.

Q. What is the solution of the Homogeneous Differential Equation? :
$\frac{dy}{dx}=\frac{x^2+y^2-xy}{x^2}$ with $y\left(1\right)=0$

Q. If $y=\frac{x}{\log |cx|}$ (where $c$ is an arbitrary constant) is the general solution of the differential equation
$\frac{dy}{dx}=\frac{y}{x}+\varphi \left(\frac{x}{y}\right)$, then the function $\varphi \left(\frac{x}{y}\right)$ is

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

Q. If $\overrightarrow{a},\overrightarrow{b}$ are collinear, then which of the following is false?

1. $\overrightarrow{a}\cdot \overrightarrow{b}=ab$
2. $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{a}$
3. $\overrightarrow{a}\cdot \overrightarrow{b}=0$
4. $\overrightarrow{a}\times \overrightarrow{b}=0$

Q. Let $f:\left(-1,1\right)\to R$ be a differential function with $f\left(0\right)=-1$ and $f^{\prime }\left(0\right)=1$. Let $g\left(x\right)={\left[f\left(2f\left(x\right)+2\right)\right]}^2$

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

Q. If $f\left(x\right)=x^3+4x^2+\lambda x+1$ is a monotonically decreasing function of $x$ in the largest possible interval $(-2,-2/3)$ then

1. $\lambda$ has no real value
2. $\lambda =2$
3. $\lambda =4$
4. $\lambda =-1$

Q. Answer the following by appropriately matching the lists based on the information given in Column $\text{I}$ and Column $\text{II}$
$\begin{array}{cc}ColumnI& ColumnII\\ \text{a.}\text{If the function}y=e^{4x}+2e^{-x}\text{is a solution of}& \\ \text{the differential equation}\frac{\frac{d^{3}y}{d{x}^3}-13\frac{dy}{dx}}{y}=K,& \\ \text{then the value of}\frac{K}{3}\text{is}& \text{p.}3\\ \text{b.}\text{Number of straight lines which satisfy the}& \\ \text{differential equation}\frac{dy}{dx}+x{\left(\frac{dy}{dx}\right)}^2-y=0& \\ \text{is}& \text{q.}4\\ \text{c.}\text{If real value of}m\text{for which the substitution,}& \\ y=u^m\text{will transform the differential equat-}& \\ \text{ion,}2x^{4}y\frac{dy}{dx}+y^4=4x^6\text{into a homogeneous}& \\ \text{equation, then the value of}2m\text{is}& \text{r.}2\\ \text{d.}\text{If the solution of differential equation}& \\ x^2\frac{d^{2}y}{d{x}^2}+2x\frac{dy}{dx}=12y\text{is}y=A{x}^m+B{x}^{-n},& \\ \text{then}|m+n|\text{is}& \text{s.}7\end{array}$

Then which of the following is correct ?

1. $a\to s,b\to r,c\to p,d\to q$
2. $a\to q,b\to r,c\to p,d\to s$
3. $a\to q,b\to p,c\to r,d\to s$
4. $a\to s,b\to q,c\to p,d\to s$

Q. If $y\left(x\right)$ is the solution of the differential equation $\frac{dy}{dx}+\left(\frac{2x+1}{x}\right)y=e^{-2x},x>0$, where $y\left(1\right)=\frac{1}{2}{e}^{-2}$, then:

1. $y\left(x\right)$ is decreasing in $(\frac{1}{2},1)$
2. $y\left(x\right)$ is decreasing in $(0,1)$
3. $y\left({\log }_{e}2\right)=\frac{{\log }_{e}2}{8}$
4. $y\left({\log }_{e}2\right)=\frac{{\log }_{e}2}{4}$

Q. Answer the following by appropriately matching the lists based on the information given in Column $\text{I}$ and Column $\text{II}$
$\begin{array}{cc}ColumnI& ColumnII\\ \text{a.}\text{If the function}y=e^{4x}+2e^{-x}\text{is a solution of}& \\ \text{the differential equation}\frac{\frac{d^{3}y}{d{x}^3}-13\frac{dy}{dx}}{y}=K,& \\ \text{then the value of}\frac{K}{3}\text{is}& \text{p.}3\\ \text{b.}\text{Number of straight lines which satisfy the}& \\ \text{differential equation}\frac{dy}{dx}+x{\left(\frac{dy}{dx}\right)}^2-y=0& \\ \text{is}& \text{q.}4\\ \text{c.}\text{If real value of}m\text{for which the substitution,}& \\ y=u^m\text{will transform the differential equat-}& \\ \text{ion,}2x^{4}y\frac{dy}{dx}+y^4=4x^6\text{into a homogeneous}& \\ \text{equation, then the value of}2m\text{is}& \text{r.}2\\ \text{d.}\text{If the solution of differential equation}& \\ x^2\frac{d^{2}y}{d{x}^2}+2x\frac{dy}{dx}=12y\text{is}y=A{x}^m+B{x}^{-n},& \\ \text{then}|m+n|\text{is}& \text{s.}7\end{array}$

Then which of the following is correct ?

1. $a\to s,b\to r,c\to p,d\to q$
2. $a\to q,b\to r,c\to p,d\to s$
3. $a\to q,b\to p,c\to r,d\to s$
4. $a\to s,b\to q,c\to p,d\to s$

Q. If $y=\frac{x}{\log |cx|}$ (where $c$ is an arbitrary constant) is the general solution of the differential equation
$\frac{dy}{dx}=\frac{y}{x}+\varphi \left(\frac{x}{y}\right)$, then the function $\varphi \left(\frac{x}{y}\right)$ is

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

Q. The differential equation for the family of curves $x^2+y^2-2ay=0$, where $a$ is an arbitrary constant is

1. $(x^2-y^2)y^{\prime }=2xy$
2. $2(x^2+y^2)y^{\prime }=xy$
3. $2\left(x^2{y}^2\right)y^{\prime }=xy$
4. $(x^2+y^2)y^{\prime }=2xy$