Logarithmic Differentiation

Q. Find the derivative of the following function:
f(x)= $\frac{1}{a{x}^2+bx+c}$

Q.

Prove the following:

$\frac{\text{tan}(\frac{\pi }{4}+x)}{\text{tan}(\frac{\pi }{4}-x)}$ = ${\left[\frac{1+\text{tan x}}{1-\text{tan x}}\right]}^2$

Q. Find the derivative of the following function:
f(x)= $(ax+b{)}^n$

Q. Let f(x) = 2x² + 4, g(x) = sin(πx), then the value of [f′(1) + g′(0)] is (where [·] represents the greatest integer function)

माना f(x) = 2x² + 4, g(x) = sin(πx), तब [f′(1) + g′(0)] का मान है (जहाँ [·] महत्तम पूर्णांक फलन को दर्शाता है

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

Q.

What is $\frac{dy}{dx}ify=(x-1{)}^2(x+2{)}^3$

1. $y[\frac{x-1}{2}+\frac{x+2}{3}]$

2. $y.\left[2\right(x-1)+3(x+2\left)\right]$

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

4. $y[\frac{2}{x-1}+\frac{3}{x+2}]$

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. Let $f$ and $g$ be two functions such that $f:[e,e^2]\to [1,2],f\left(x\right)=\ln x$ and $g:[1,2]\to [e,e^2],g\left(x\right)=e^x.$ Then, $fog$ is

1. neither one-one nor onto function.
2. onto but not one-one function.
3. one-one and onto function.
4. one-one but not onto function.

Q. If $y=\frac{a{x}^2}{(x-a)(x-b)(x-c)}+\frac{bx}{(x-b)(x-c)}+\frac{c}{x-c}+1,$ then $\frac{y^{\prime }}{y}$ is equal to
$(\text{Here},y^{\prime }=\frac{dy}{dx})$

1. $\frac{1}{x}(\frac{a}{a-x}+\frac{b}{b-x}+\frac{c}{c-x})$
2. $\frac{1}{x}(\frac{b}{a-x}+\frac{c}{b-x}+\frac{a}{c-x})$
3. $\frac{abc}{x}(\frac{a}{a-x}+\frac{b}{b-x}+\frac{c}{c-x})$
4. $\frac{1}{x}(\frac{bc}{a-x}+\frac{ac}{b-x}+\frac{ab}{c-x})$

Q. If $f\left(x\right)={(25-x^4)}^{1/4},0<x<\sqrt{5}$, then $f\left(f\left(\frac{1}{3}\right)\right)=$

1. $\frac{1}{3}$
2. $3^{1/3}$
3. $27$
4. $\frac{2}{3}$

Q. Find the derivative of the function $f\left(x\right)=x^x$

1. $f\left(x\right)\cdot [1–\ln x]$
2. $f\left(x\right)\cdot [1+\ln x]$
3. $f\left(x\right)\cdot \ln x$
4. $f\left(x\right)-\ln x$

Q. $If{x}^{cosy}+y^{cosx}=5.$ Then

1. at x = 0, y = 0, y’ = 0
2. at x = 0, y = 1, y’ = 0
3. at x = y = 1, y’ = – 1
4. at x = 1, y = 0, y’ = 1

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

1. $(1+\ln x{)}^{-1}$
2. $(1+\ln x{)}^{-2}$
3. $\ln x(1+\ln x{)}^{-2}$
4. None of these