Logarithmic Differentiation

Q.

What are the Logarithm rules?

Q. The domain of the function $f\left(x\right)={\sin }^{-1}\left(\frac{3x^2+x-1}{(x-1{)}^2}\right)+{\cos }^{-1}\left(\frac{x-1}{x+1}\right)$ is

1. $[0,\frac{1}{2}]$
2. $[\frac{1}{4},\frac{1}{2}]\cup \left\{0\right\}$
3. $[0,\frac{1}{4}]$
4. $[-2,0]\cup [\frac{1}{4},\frac{1}{2}]$

Q.

If $x^m.y^n=(x+y{)}^{m+n}$, then $\frac{dy}{dx}$ is

1. $\frac{y}{x}$

2. $\frac{x+y}{xy}$

3. $xy$

4. $\frac{x}{y}$

Q. for $x>1$, If $(2x{)}^{2y}=4e^{2x-2y}$, then
$(1+{\log }_{e}2x{)}^2\frac{dy}{dx}$ is equal to :

1. $\frac{x{\log }_{e}2x+{\log }_{e}2}{x}$
2. ${\log }_{e}2x$
3. $\frac{x{\log }_{e}2x-{\log }_{e}2}{x}$
4. $x{\log }_{e}2x$

Q. for $x>1$, If $(2x{)}^{2y}=4e^{2x-2y}$, then
$(1+{\log }_{e}2x{)}^2\frac{dy}{dx}$ is equal to :

1. $\frac{x{\log }_{e}2x+{\log }_{e}2}{x}$
2. ${\log }_{e}2x$
3. $\frac{x{\log }_{e}2x-{\log }_{e}2}{x}$
4. $x{\log }_{e}2x$

Q.

If $x^m.y^n=(x+y{)}^{m+n}$, then $\frac{dy}{dx}$ is

1. $\frac{y}{x}$

2. $\frac{x+y}{xy}$

3. $xy$

4. $\frac{x}{y}$

Q. If $x=y^{x^{y^x.......\infty }}\text{then}\frac{dy}{dx}\text{at}x=1$ is

1. 1
2. 2
3. 5
4. 7

Q.

If $x^m.y^n=(x+y{)}^{m+n}$, then $\frac{dy}{dx}$ is

1. $\frac{y}{x}$

2. $\frac{x+y}{xy}$

3. $xy$

4. $\frac{x}{y}$

Q. The solution of the differential equation $ydx-xdy=y^2\tan \left(\frac{x}{y}\right)dx$ is
( $C$ is constant of integration)

1. $\frac{x}{y}=C{e}^x$
2. $\sin \left(\frac{x}{y}\right)=C{e}^x$
3. $\cos \left(\frac{x}{y}\right)=C{e}^x$
4. $x=Cy$

Q. If $y=x^{e^x}$, then $\frac{dy}{dx}=$

1. $y(logx+e^x)$
2. $ylogx(\frac{1}{2}+e^x)$
3. $y{e}^x(logx+x)$
4. $y{e}^x(logx+\frac{1}{x})$

Q. If $x=e^{y+e^{y+e^{y+e^{y+\cdots \infty }}}}$

1. $\frac{1}{x}$
2. $\frac{1-x}{x}$
3. $\frac{x}{1+x}$
4. $Noneofthese$