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
Differentiate the following functions from first principles e−x.
Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?
Differentiate \[\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}\] ?
Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?
Differentiate \[\log \left( \cos x^2 \right)\] ?
If \[y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}\] prove that \[\frac{dy}{dx} = \sec 2x\] ?
If \[y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}\] , prove that \[\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}\] ?
If \[y = x \sin^{- 1} x + \sqrt{1 - x^2}\] ,prove that \[\frac{dy}{dx} = \sin^{- 1} x\] ?
If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?
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 \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
If \[y = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x > 0\] ,prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ?
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: \[y^3 - 3x y^2 = x^3 + 3 x^2 y\] ?
If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[e^{x \log x}\] ?
Differentiate \[\sin \left( x^x \right)\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\log x }+ \left( \log x \right)^x\] ?
If \[x^y + y^x = \left( x + y \right)^{x + y} , \text{ find } \frac{dy}{dx}\] ?
If \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?
If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?
Write the derivative of sinx with respect to cos x ?
Differentiate x2 with respect to x3
Differentiate \[\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text { if }0 < x < 1\] ?
Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and `f' (3) = 9`, write the value of `g' (9)`.
If \[\frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \text { and y } = \sin^{- 1} \left( \sin x \right), \text { find } \frac{dy}{dx} \] ?
If \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?
Differential coefficient of sec(tan−1 x) is ______.
If \[\sqrt{1 - x^6} + \sqrt{1 - y^6} = a^3 \left( x^3 - y^3 \right)\] then \[\frac{dy}{dx}\] is equal to ____________ .
Find the second order derivatives of the following function tan−1 x ?
If y = ex cos x, prove that \[\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)\] ?
If x = 4z2 + 5, y = 6z2 + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?
If y = a + bx2, a, b arbitrary constants, then
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume, and radius r.
