Log Function

Q.

Using first principle find derivative of log ax+b

Q.

If the lines $ax+ky+10=0,bx+(k+1)y+10=0$ and $cx+(k+2)y+10=0$ are concurrent, then:

1. $a,b,c$ are in G.P.

2. $a,b,c$ are in H.P.

3. $a,b,c$ are in A.P.

4. ${(a+1)}^2=c$

Q.

If $x$ varies inversely as $y$ and $x=4$ when $y=6$, then when $x=3$ the value of $y$ is _______.

Q. $\mathrm{If}y=1+\frac{\alpha }{\left({\displaystyle \frac{1}{x}-\alpha }\right)}+\frac{{\displaystyle \raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$x$}\right.}}{\left({\displaystyle \frac{1}{x}-\alpha }\right)\left({\displaystyle \frac{1}{x}-\beta }\right)}+\frac{{\displaystyle \raisebox{1ex}{$\gamma $}\!\left/ \!\raisebox{-1ex}{$x^2$}\right.}}{\left({\displaystyle \frac{1}{x}-\alpha }\right)\left({\displaystyle \frac{1}{x}-\beta }\right)\left({\displaystyle \frac{1}{x}-\gamma }\right)},\mathrm{find}\frac{dy}{dx}.$

Q. If $y=x^{(\log x{)}^{\log \left(\log x\right)}}$, then $\frac{dy}{dx}$ is

1. $\frac{y}{x}[\ln x^{\ln x-1}+2\ln x\ln (\ln x\left)\right]$
2. $\frac{y}{x\ln x}\left[\right(\ln x{)}^2+2\ln \left(\ln x\right)]$
3. $\frac{y}{x}\left(\log x{)}^{\log \left(\log x\right)}\right(2\log \left(\log x\right)+1)$
4. $\frac{y\log y}{x\log x}\left[2\log \right(\log x)+1]$

Q. If $x^y=e^{x-y},\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\mathrm{dy}}{\mathrm{dx}}\mathrm{at}x=e$ is

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

Q. Match the following functions satisfying particular functional relationship

	Column-I
(A)	$f(x+y)=f\left(x\right)+f\left(y\right)$
(B)	$f\left(xy\right)=f\left(x\right)+f\left(y\right)$
(C)	$f(x+y)=f\left(x\right)\cdot f\left(y\right)$
(D)	$f\left(X\right)+f\left(y\right)=f\left(\frac{x+y}{1-xy}\right)$

	Column-II
(p)	${\log }_{3}x$
(q)	$ta{n}^{-1}x$
(r)	$3x$
(s)	$3^x$
(t)	A transcendental function

Q. If $f\left(x\right)={\log }_x\left(\ln x\right)$, then

1. $f^{\prime }\left(e\right)=e$
2. $f^{\prime }\left(e\right)=\frac{1}{e}$
3. $f^{\prime }\left(e^2\right)=\frac{1}{e^2}$
4. $f^{\prime }\left(e^2\right)=\frac{(1-\ln 2)}{(2e{)}^2}$

Q. If $y={\left(1+\frac{1}{x}\right)}^x,\mathrm{then}\frac{dy}{dx}=$

(a) ${\left(1+\frac{1}{x}\right)}^x\left(1+\frac{1}{x}\right)-\frac{1}{x+1}$

(b) ${\left(1+\frac{1}{x}\right)}^x\log \left(1+\frac{1}{x}\right)$

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

(d) ${\left(x+\frac{1}{x}\right)}^x\left\{\log \left(1+\frac{1}{x}\right)+\frac{1}{x+1}\right\}$

Q. 110. Find dy/dx if (x+y) = xy

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

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

Q. If $y=\left|{\log }_{e}x\right|$, find $\frac{d^{2}y}{d{x}^2}$.

Q. If $xy=e^{x-y},\mathrm{find}\frac{dy}{dx}$.

Q. If $y^x+x^y+x^x=a^b$, find $\frac{dy}{dx}$.

Q. Find $\frac{dy}{dx}$ of $y^x=x^y$

Q. $\int \left(\frac{1}{\log x}-\frac{1}{{\left(\log x\right)}^2}\right)dx$

Q. (1 + x²) dy = xy dx

Q. If $y=x^{(\log x{)}^{\log \left(\log x\right)}}$, then $\frac{dy}{dx}$ is

1. $\frac{y}{x}[\ln x^{\ln x-1}+2\ln x\ln (\ln x\left)\right]$
2. $\frac{y}{x\ln x}\left[\right(\ln x{)}^2+2\ln \left(\ln x\right)]$
3. $\frac{y}{x}\left(\log x{)}^{\log \left(\log x\right)}\right(2\log \left(\log x\right)+1)$
4. $\frac{y\log y}{x\log x}\left[2\log \right(\log x)+1]$

Q. If $f\left(x\right)={\log }_x\left(\ln x\right)$, then

1. $f^{\prime }\left(e\right)=e$
2. $f^{\prime }\left(e\right)=\frac{1}{e}$
3. $f^{\prime }\left(e^2\right)=\frac{1}{e^2}$
4. $f^{\prime }\left(e^2\right)=\frac{(1-\ln 2)}{(2e{)}^2}$

Q. $\frac{dr}{dt}=-rt,r\left(0\right)=r_0$

Q. $\int \left(x+1\right)e^x\log \left(x{e}^x\right)dx$

Q. If $l=\log \frac{a^2}{bc}$, $m=\log \frac{b^2}{ca}$ and $n=\log \frac{c^2}{ab}$, find the value of $l+m+n$

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

Q. If f(x)=cos(logx) then f(x)f(y)-1/2[f(x/y)+f(xy)] is equal to?

Q. If $y=\left(x-1\right)\log \left(x-1\right)-\left(x+1\right)\log \left(x+1\right)$, prove that $\frac{dy}{dc}=\log \left(\frac{x-1}{1+x}\right)$

Q. If $f\left(x\right)={\log }_x\left(\ln x\right)$, then

1. $f^{\prime }\left(e\right)=e$
2. $f^{\prime }\left(e\right)=\frac{1}{e}$
3. $f^{\prime }\left(e^2\right)=\frac{1}{e^2}$
4. $f^{\prime }\left(e^2\right)=\frac{(1-\ln 2)}{(2e{)}^2}$

Q. If $f\left(x\right)={\log }_x\left(\ln x\right)$, then

1. $f^{\prime }\left(e\right)=e$
2. $f^{\prime }\left(e\right)=\frac{1}{e}$
3. $f^{\prime }\left(e^2\right)=\frac{1}{e^2}$
4. $f^{\prime }\left(e^2\right)=\frac{(1-\ln 2)}{(2e{)}^2}$

Q. If $x^y=e^{x-y},\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\mathrm{dy}}{\mathrm{dx}}\mathrm{at}x=e$ is

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

Q. If $y=\sqrt{x+1}+\sqrt{x-1}$, prove that $\sqrt{x^2-1}\frac{dy}{dx}=\frac{1}{2}y$

Q. If $y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+..\mathrm{to}\infty }}}$, prove that $\left(2y-1\right)\frac{dy}{dx}=\frac{1}{x}$

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