Advertisements
Advertisements
प्रश्न
If \[y = \sec^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right)\] then write the value of \[\frac{dy}{dx} \] ?
Advertisements
उत्तर
\[\text{ We have, y } = \sec^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right) \]
\[ \Rightarrow y = \cos^{- 1} \left( \frac{x - 1}{x + 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right) ........\left[ \because \sec^{- 1} x = \cos^{- 1} \left( \frac{1}{x} \right) \right]\]
\[ \Rightarrow y = \frac{\pi}{2} ..........\left[ \because \sin^{- 1} x + \cos^{- 1} x = \frac{\pi}{2} \right]\]
\[\Rightarrow \frac{dy}{dx} = 0\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following function from first principles \[e^\sqrt{\cot x}\] .
Differentiate sin (3x + 5) ?
Differentiate \[\sin^{- 1} \left( \frac{x}{\sqrt{x^2 + a^2}} \right)\] ?
\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sin x}{1 + \cos x} \right), - \pi < x < \pi\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Differentiate
\[\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
If \[y = \cot^{- 1} \left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}\], show that \[\frac{dy}{dx}\] is independent of x. ?
If \[y = se c^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right), x > 0 . \text{ Find} \frac{dy}{dx}\] ?
If \[y = \cos^{- 1} \left\{ \frac{2x - 3 \sqrt{1 - x^2}}{\sqrt{13}} \right\}, \text{ find } \frac{dy}{dx}\] ?
If \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\] \[\frac{dy}{dx} \] ?
Find \[\frac{dy}{dx}\] in the following case \[\tan^{- 1} \left( x^2 + y^2 \right) = a\] ?
If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?
Differentiate \[\left( \sin^{- 1} x \right)^x\] ?
Differentiate \[\left( x^x \right) \sqrt{x}\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = e^x + {10}^x + x^x\] ?
find \[\frac{dy}{dx}\] \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?
If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?
If \[e^x + e^y = e^{x + y}\] , prove that
\[\frac{dy}{dx} + e^{y - x} = 0\] ?
If \[y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + . . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sin x}{1 - 2 y}\] ?
If \[y = \left( \cos x \right)^{\left( \cos x \right)^{\left( \cos x \right) . . . \infty}}\],prove that \[\frac{dy}{dx} = - \frac{y^2 \tan x}{\left( 1 - y \log \cos x \right)}\]?
Differentiate log (1 + x2) with respect to tan−1 x ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?
If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ?
\[\frac{d}{dx} \left\{ \tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right) \right\} \text { equals }\] ______________ .
The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to \[\cos^{- 1} x\] is ___________ .
Find the second order derivatives of the following function log (log x) ?
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \cdot \frac{dy}{dx} + y \cos^2 x = 0\] ?
If \[y = e^{2x} \left( ax + b \right)\] show that \[y_2 - 4 y_1 + 4y = 0\] ?
If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find \[\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}\] ?
\[ \text { If x } = a \sin t \text { and y } = a\left( \cos t + \log \tan\frac{t}{2} \right), \text { find } \frac{d^2 y}{d x^2} \] ?
If x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?
If y = xn−1 log x then x2 y2 + (3 − 2n) xy1 is equal to
