Advertisements
Advertisements
प्रश्न
If \[y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)\], then \[\frac{d^2 y}{d x^2} =\]
विकल्प
2
1
0
−1
Advertisements
उत्तर
(c) 0
\[y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left\{ \frac{1 - 2 \log_e x}{1 + 2 \log_e x} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left( \frac{\frac{1 - 2 \log_e x}{1 + 2 \log_e x} + \frac{3 + 2 \log_e x}{1 - 6 \log_e x}}{1 - \left( \frac{1 - 2 \log_e x}{1 + 2 \log_e x} \right)\left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left\{ \frac{\left( 1 - 2 \log_e x \right)\left( 1 - 6 \log_e x \right) + \left( 3 + 2 \log_e x \right)\left( 1 + 2 \log_e x \right)}{\left( 1 + 2 \log_e x \right)\left( 1 - 6 \log_e x \right) - \left( 1 - 2 \log_e x \right)\left( 3 + 2 \log_e x \right)} \right\}\]
\[ \Rightarrow y = \tan^{- 1} \left\{ \frac{1 - 8 \log_e x + 12 \left( \log_e x \right)^2 + 3 + 8 \log_e x + 4 \left( \log_e x \right)^2}{1 - 4 \log_e x - 12 \left( \log_e x \right)^2 - 3 + 4 \log_e x + 4 \left( \log_e x \right)^2} \right\}\]
\[ \Rightarrow y = \tan^{- 1} \left\{ \frac{1 - 8 \log_e x + 12 \left( \log_e x \right)^2 + 3 + 8 \log_e x + 4 \left( \log_e x \right)^2}{1 - 4 \log_e x - 12 \left( \log_e x \right)^2 - 3 + 4 \log_e x + 4 \left( \log_e x \right)^2} \right\}\]
\[ \Rightarrow y = \tan^{- 1} \left\{ \frac{4 + 16 \left( \log_e x \right)^2}{- 2 - 8 \left( \log_e x \right)^2} \right\}\]
\[ \Rightarrow y = \tan^{- 1} \left[ \frac{4\left\{ 1 + 4 \left( \log_e x \right)^2 \right\}}{- 2\left\{ 1 + 4 \left( \log_e x \right)^2 \right\}} \right]\]
\[ \Rightarrow y = \tan^{- 1} \left[ - 2 \right]\]
\[ \Rightarrow \frac{dy}{dx} = 0\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = 0\]
APPEARS IN
संबंधित प्रश्न
If the function f(x)=2x3−9mx2+12m2x+1, where m>0 attains its maximum and minimum at p and q respectively such that p2=q, then find the value of m.
If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.
Differentiate `2^(x^3)` ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\]?
If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?
If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 \text{cosec }2x \] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{2} \right\}, 0 < x < 1\] ?
If \[y = \cos^{- 1} \left\{ \frac{2x - 3 \sqrt{1 - x^2}}{\sqrt{13}} \right\}, \text{ find } \frac{dy}{dx}\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{2^{x + 1} \cdot 3^x}{1 + \left(36 \right)^x} \right\}\] with respect to x.
Find \[\frac{dy}{dx}\] in the following case \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?
Find \[\frac{dy}{dx}\] in the following case \[e^{x - y} = \log \left( \frac{x}{y} \right)\] ?
If \[y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}\] ?
If \[\sqrt{y + x} + \sqrt{y - x} = c, \text {show that } \frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}\] ?
Differentiate \[x^{\sin x}\] ?
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Differentiate \[\left( \log x \right)^{ \log x }\] ?
Differentiate \[x^{x^2 - 3} + \left( x - 3 \right)^{x^2}\] ?
Find \[\frac{dy}{dx}\] \[y = e^{3x} \sin 4x \cdot 2^x\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)\] ?
If \[x^y \cdot y^x = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( y + x \log y \right)}{x \left( y \log x + x \right)}\] ?
If \[e^{x + y} - x = 0\] ,prove that \[\frac{dy}{dx} = \frac{1 - x}{x}\] ?
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
If \[y = \log\frac{x^2 + x + 1}{x^2 - x + 1} + \frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{\sqrt{3} x}{1 - x^2} \right), \text{ find } \frac{dy}{dx} .\] ?
Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( \frac{1}{2 \sqrt{2}}, \frac{1}{2} \right)\] ?
If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .
If \[f\left( x \right) = \sqrt{x^2 - 10x + 25}\] then the derivative of f (x) in the interval [0, 7] is ____________ .
Find the second order derivatives of the following function tan−1 x ?
If y = e−x cos x, show that \[\frac{d^2 y}{d x^2} = 2 e^{- x} \sin x\] ?
Find \[\frac{d^2 y}{d x^2}\] where \[y = \log \left( \frac{x^2}{e^2} \right)\] ?
If x = 4z2 + 5, y = 6z2 + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?
If y = cosec−1 x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If y = a + bx2, a, b arbitrary constants, then
f(x) = 3x2 + 6x + 8, x ∈ R
