Advertisements
Advertisements
प्रश्न
If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 \text{cosec }2x \] ?
Advertisements
उत्तर
\[\text{ We have, y} = \frac{1}{2}\log\left( \frac{1 - \cos2x}{1 + \cos2x} \right)\]
\[ \Rightarrow y = \frac{1}{2}\log\left( \frac{2 \sin^2 x}{2 \cos^2 x} \right) \]
\[ \Rightarrow y = \frac{1}{2}\log\left( \tan^2 x \right)\]
\[ \Rightarrow y = \frac{2}{2}\log \tan x \]
\[ \Rightarrow y = \log \tan x\]
Differentiate with respect to x,
\[\frac{d y}{d x} = \left( \log \tan x \right)\]
\[ = \frac{1}{\tan x} \times \frac{d}{dx}\left( \tan x \right) \]
\[ = \frac{\sec^2 x}{\tan x}\]
\[ = \frac{1}{\cos^2 x \times \frac{\sin x}{\cos x}}\]
\[ = \frac{1}{\sin x \cos x}\]
\[ = \frac{2}{2\sin x \cos x}\]
\[ = \frac{2}{\sin2x} \]
\[So, \frac{d y}{d x} = 2\text{cosec }2x\]
APPEARS IN
संबंधित प्रश्न
Prove that `y=(4sintheta)/(2+costheta)-theta `
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 the following functions from first principles log cosec x ?
Differentiate log7 (2x − 3) ?
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?
If \[y = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] .prove that \[\frac{dy}{dx} = 1 - y^2\] ?
If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?
Differentiate \[\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
If \[y = \cos^{- 1} \left\{ \frac{2x - 3 \sqrt{1 - x^2}}{\sqrt{13}} \right\}, \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[{10}^{ \log \sin x }\] ?
Differentiate \[{10}^\left( {10}^x \right)\] ?
If \[y = \sin \left( x^x \right)\] prove that \[\frac{dy}{dx} = \cos \left( x^x \right) \cdot x^x \left( 1 + \log x \right)\] ?
If \[xy = e^{x - y} , \text{ find } \frac{dy}{dx}\] ?
If \[\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}\] ?
If \[x = a \left( \frac{1 + t^2}{1 - t^2} \right) \text { and y } = \frac{2t}{1 - t^2}, \text { find } \frac{dy}{dx}\] ?
If \[x = 10 \left( t - \sin t \right), y = 12 \left( 1 - \cos t \right), \text { find } \frac{dy}{dx} .\] ?
If \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?
If \[f'\left( x \right) = \sqrt{2 x^2 - 1} \text { and y } = f \left( x^2 \right)\] then find \[\frac{dy}{dx} \text { at } x = 1\] ?
If \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?
If \[f\left( x \right) = \tan^{- 1} \sqrt{\frac{1 + \sin x}{1 - \sin x}}, 0 \leq x \leq \pi/2, \text{ then } f' \left( \pi/6 \right) \text{ is }\] _________ .
Given \[f\left( x \right) = 4 x^8 , \text { then }\] _________________ .
The derivative of \[\sec^{- 1} \left( \frac{1}{2 x^2 + 1} \right) \text { w . r . t }. \sqrt{1 + 3 x} \text { at } x = - 1/3\]
Let \[\cup = \sin^{- 1} \left( \frac{2 x}{1 + x^2} \right) \text { and }V = \tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text { then } \frac{d \cup}{dV} =\] ____________ .
If \[y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =\] __________ .
If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .
If \[y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] __________ .
Find the second order derivatives of the following function x cos x ?
If \[y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2\] show that \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2\] ?
If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?
If x = t2 and y = t3, find \[\frac{d^2 y}{d x^2}\] ?
If \[\frac{d}{dx}\left[ x^n - a_1 x^{n - 1} + a_2 x^{n - 2} + . . . + \left( - 1 \right)^n a_n \right] e^x = x^n e^x\] then the value of ar, 0 < r ≤ n, is equal to
If logy = tan–1 x, then show that `(1+x^2) (d^2y)/(dx^2) + (2x - 1) dy/dx = 0 .`
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
