English

If Y = 1 2 Log ( 1 − Cos 2 X 1 + Cos 2 X ) , Prove Tha D Y D X = 2 C O S E C 2 X ? - Mathematics

Advertisements
Advertisements

Question

If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 \text{cosec }2x \] ?

Sum
Advertisements

Solution

\[\text{ We have, y} = \frac{1}{2}\log\left( \frac{1 - \cos2x}{1 + \cos2x} \right)\]

\[ \Rightarrow y = \frac{1}{2}\log\left( \frac{2 \sin^2 x}{2 \cos^2 x} \right) \]

\[ \Rightarrow y = \frac{1}{2}\log\left( \tan^2 x \right)\]

\[ \Rightarrow y = \frac{2}{2}\log \tan x \]

\[ \Rightarrow y = \log \tan x\]

Differentiate with respect to x,

\[\frac{d y}{d x} = \left( \log \tan x \right)\]

\[ = \frac{1}{\tan x} \times \frac{d}{dx}\left( \tan x \right) \]

\[ = \frac{\sec^2 x}{\tan x}\]

\[ = \frac{1}{\cos^2 x \times \frac{\sin x}{\cos x}}\]

\[ = \frac{1}{\sin x \cos x}\]

\[ = \frac{2}{2\sin x \cos x}\]

\[ = \frac{2}{\sin2x} \]

\[So, \frac{d y}{d x} = 2\text{cosec }2x\]

shaalaa.com
  Is there an error in this question or solution?
Chapter 11: Differentiation - Exercise 11.02 [Page 38]

APPEARS IN

RD Sharma Mathematics [English] Class 12
Chapter 11 Differentiation
Exercise 11.02 | Q 68 | Page 38

Video TutorialsVIEW ALL [1]

RELATED QUESTIONS

Differentiate the following functions from first principles x2ex ?


Differentiate tan 5x° ?


Differentiate \[\log \left( cosec x - \cot x \right)\] ?


Differentiate  \[e^x \log \sin 2x\] ?


Differentiate \[\cos \left( \log x \right)^2\] ?


Differentiate \[\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1\] ?


Differentiate  \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\]  ?


Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?


Differentiate the following with respect to x

\[\cos^{- 1} \left( \sin x \right)\]


If \[\sqrt{1 - x^2} + \sqrt{1 - y^2} = a \left( x - y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{1 - x^2}\] ?


Differentiate \[x^{\sin x}\]  ?


Differentiate \[x^{\cos^{- 1} x}\] ?


Differentiate \[{10}^{ \log \sin x }\] ?


Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?


Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?


Find \[\frac{dy}{dx}\]  \[y = x^n + n^x + x^x + n^n\] ?

If \[x^y + y^x = \left( x + y \right)^{x + y} , \text{ find } \frac{dy}{dx}\] ?


If \[y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}\], prove that \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?

 


Find \[\frac{dy}{dx}\] , when \[x = \frac{3 at}{1 + t^2}, \text{ and } y = \frac{3 a t^2}{1 + t^2}\] ?


Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?


If  \[x = \frac{\sin^3 t}{\sqrt{\cos 2 t}}, y = \frac{\cos^3 t}{\sqrt{\cos t 2 t}}\] , find\[\frac{dy}{dx}\] ?

 


If  \[x = a\sin2t\left( 1 + \cos2t \right) \text { and y } = b\cos2t\left( 1 - \cos2t \right)\] , show that at  \[t = \frac{\pi}{4}, \frac{dy}{dx} = \frac{b}{a}\] ?


\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?


Write the derivative of sinx with respect to cos x ?


Differentiate  \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\]\[x \in \left( 0, 1 \right)\]  ?

 


Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with  respect to \[\sec^{- 1} x\] ?


If \[f\left( 1 \right) = 4, f'\left( 1 \right) = 2\] find the value of the derivative of  \[\log \left( f\left( e^x \right) \right)\] w.r. to x at the point x = 0 ?

 


Given  \[f\left( x \right) = 4 x^8 , \text { then }\] _________________ .


\[\frac{d}{dx} \left\{ \tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right) \right\} \text { equals }\] ______________ .


If \[y = \frac{1}{1 + x^{a - b} +^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} + \frac{1}{1 + x^{b - a} + x^{c - a}}\] then \[\frac{dy}{dx}\]  is equal to ______________ .


Find the second order derivatives of the following function x3 log ?


If x = a(1 − cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{1}{a}\text { at } \theta = \frac{\pi}{2}\] ?


If y = (sin−1 x)2, prove that (1 − x2)

\[\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?


If y = cos−1 x, find \[\frac{d^2 y}{d x^2}\] in terms of y alone ?


If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?


\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?


\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]

 


If \[y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =\] ?


If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]


Differentiate the following with respect to x

\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×