Advertisements
Advertisements
Question
If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
Advertisements
Solution
\[\text{ We have }, \sec\left( \frac{x + y}{x - y} \right) = a\]
\[ \Rightarrow \frac{x + y}{x - y} = \sec^{- 1} \left( a \right)\]
Differentiate with respect to x, we get,
\[\Rightarrow \left[ \frac{\left( x - y \right)\frac{d}{dx}\left( x + y \right) - \left( x + y \right)\frac{d}{dx}\left( x - y \right)}{\left( x - y \right)^2} \right] = 0\]
\[ \Rightarrow \left( x - y \right) \left( 1 + \frac{d y}{d x} \right) - \left( x + y \right) \left( 1 - \frac{d y}{d x} \right) = 0\]
\[ \Rightarrow \left( x - y \right) + \left( x - y \right)\frac{d y}{d x} - \left( x + y \right) + \left( x + y \right)\frac{d y}{d x} = 0\]
\[ \Rightarrow \frac{d y}{d x}\left[ x - y + x + y \right] = x + y - x + y\]
\[ \Rightarrow \frac{d y}{d x}\left( 2x \right) = 2y\]
\[ \Rightarrow \frac{d y}{d x} = \frac{y}{x}\]
APPEARS IN
RELATED QUESTIONS
Prove that `y=(4sintheta)/(2+costheta)-theta `
Differentiate the following functions from first principles \[e^\sqrt{2x}\].
Differentiate sin (3x + 5) ?
Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?
Differentiate \[e^{\sin^{- 1} 2x}\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
If \[y = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] .prove that \[\frac{dy}{dx} = 1 - y^2\] ?
If \[y = e^x + e^{- x}\] prove that \[\frac{dy}{dx} = \sqrt{y^2 - 4}\] ?
Differentiate \[\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1\] ?
If \[y = se c^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right), x > 0 . \text{ Find} \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?
If \[\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)\] Prove that \[\frac{dy}{dx} = \frac{x + y}{x - y}\] ?
If \[\sin^2 y + \cos xy = k,\] find \[\frac{dy}{dx}\] at \[x = 1 , \] \[y = \frac{\pi}{4} .\]
Differentiate \[x^{\sin x}\] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Differentiate \[\sin \left( x^x \right)\] ?
Differentiate \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = e^x + {10}^x + x^x\] ?
If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
If \[e^{x + y} - x = 0\] ,prove that \[\frac{dy}{dx} = \frac{1 - x}{x}\] ?
If \[y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}\], prove that \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?
If f (x) is an even function, then write whether `f' (x)` is even or odd ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
The derivative of the function \[\cot^{- 1} \left| \left( \cos 2 x \right)^{1/2} \right| \text{ at } x = \pi/6 \text{ is }\] ______ .
If \[f\left( x \right) = \left| x^2 - 9x + 20 \right|\] then `f' (x)` is equal to ____________ .
If \[f\left( x \right) = \left| x - 3 \right| \text { and }g\left( x \right) = fof \left( x \right)\] is equal to __________ .
Find the second order derivatives of the following function tan−1 x ?
If x = sin t, y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?
If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find \[\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}\] ?
\[\text { If }y = A e^{- kt} \cos\left( pt + c \right), \text { prove that } \frac{d^2 y}{d t^2} + 2k\frac{d y}{d t} + n^2 y = 0, \text { where } n^2 = p^2 + k^2 \] ?
If \[y = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}\] .....to ∞, then write \[\frac{d^2 y}{d x^2}\] in terms of y ?
If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =
If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
