Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\text{ We have, x } = \sin^{- 1} \left( \frac{2t}{1 + t^2} \right)\]
\[\text {Put t } = \tan\theta\]
\[ \Rightarrow - 1 < \tan\theta < 1\]
\[ \Rightarrow - \frac{\pi}{4} < \theta < \frac{\pi}{4}\]
\[ \Rightarrow - \frac{\pi}{2} < 2\theta < \frac{\pi}{2}\]
\[ \therefore x = \sin^{- 1} \left( \frac{2 \tan\theta}{1 + \tan^2 \theta} \right)\]\[ \Rightarrow x = \sin^{- 1} \left( \sin2\theta \right)\]
\[ \Rightarrow x = 2\theta .......\left[ \because - \frac{\pi}{2} < 2\theta < \frac{\pi}{2} \right]\]
\[ \Rightarrow x = 2\left( \tan^{- 1} t \right) .........\left[ \because t = \sin\theta \right]\]
\[\Rightarrow \frac{dx}{dt} = \frac{2}{1 + t^2} . . . \left( i \right)\]
\[\text { Now, y } = \tan^{- 1} \left( \frac{2t}{1 - t^2} \right)\]
\[\text { put t } = \tan\theta\]
\[ \Rightarrow y = \tan^{- 1} \left( \frac{2 \tan\theta}{1 - \tan^2 \theta} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left( \tan 2\theta \right) \]
\[ \Rightarrow y = 2\theta .......\left[ \because - \frac{\pi}{2} < 2\theta < \frac{\pi}{2} \right]\]
\[ \Rightarrow y = 2 \tan^{- 1} t .....\left[ \because t = \tan\theta \right]\]
\[\Rightarrow \frac{dy}{dt} = \frac{2}{1 + t^2} . . . \left( ii \right)\]
\[\text { Dividing equation } \left( ii \right) \text { by } \left( i \right), \]
\[\frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2}{1 + t^2} \times \frac{1 + t^2}{2}\]
\[ \Rightarrow \frac{dy}{dx} = 1\]
APPEARS IN
संबंधित प्रश्न
If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Prove that `y=(4sintheta)/(2+costheta)-theta `
Differentiate the following functions from first principles e−x.
Differentiate tan (x° + 45°) ?
Differentiate sin2 (2x + 1) ?
Differentiate tan 5x° ?
Differentiate \[e^\sqrt{\cot x}\] ?
Differentiate \[\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)\] ?
Differentiate \[\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}\] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
If \[y = \sqrt{x^2 + a^2}\] prove that \[y\frac{dy}{dx} - x = 0\] ?
Prove that \[\frac{d}{dx} \left\{ \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{- 1} \frac{x}{a} \right\} = \sqrt{a^2 - x^2}\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{5 x}{1 - 6 x^2} \right), - \frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}}\] ?
Differentiate
\[\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
If \[y = \sin^{- 1} \left( \frac{x}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right), 0 < x < \infty\] prove that \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?
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 \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, 0 < x < \frac{1}{2}, \text{ find } \frac{dy}{dx} .\] ?
If \[\sqrt{1 - x^2} + \sqrt{1 - y^2} = a \left( x - y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{1 - x^2}\] ?
If `ysqrt(1-x^2) + xsqrt(1-y^2) = 1` prove that `dy/dx = -sqrt((1-y^2)/(1-x^2))`
If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
Differentiate \[x^{\sin^{- 1} x}\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?
Find \[\frac{dy}{dx}\], when \[x = a t^2 \text{ and } y = 2\ at \] ?
If \[\frac{dy}{dx}\] when \[x = a \cos \theta \text{ and } y = b \sin \theta\] ?
Find \[\frac{dy}{dx}\] when \[x = \frac{2 t}{1 + t^2} \text{ and } y = \frac{1 - t^2}{1 + t^2}\] ?
Find \[\frac{dy}{dx}\] , when \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?
If \[x = 2 \cos \theta - \cos 2 \theta \text{ and y} = 2 \sin \theta - \sin 2 \theta\], prove that \[\frac{dy}{dx} = \tan \left( \frac{3 \theta}{2} \right)\] ?
Differentiate x2 with respect to x3
Differentiate \[\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text { if }0 < x < 1\] ?
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 \[\frac{dy}{dx} \] ?
If f (x) is an even function, then write whether `f' (x)` is even or odd ?
If y = 500 e7x + 600 e−7x, show that \[\frac{d^2 y}{d x^2} = 49y\] ?
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
If y = xx, prove that \[\frac{d^2 y}{d x^2} - \frac{1}{y} \left( \frac{dy}{dx} \right)^2 - \frac{y}{x} = 0 .\]
f(x) = xx has a stationary point at ______.
