Advertisements
Advertisements
प्रश्न
Differentiate \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right)\] with respect to \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right), \text { if } - \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
उत्तर
\[\text { Let, u } = \tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right)\]
\[\text { Put x } = \sin\theta\]
\[ \Rightarrow \theta = \sin^{- 1} x\]
\[ \Rightarrow u = \tan^{- 1} \left( \frac{\sin\theta}{\sqrt{1 - \sin^2 \theta}} \right) \]
\[ \Rightarrow u = \tan^{- 1} \left( \frac{\sin\theta}{\cos\theta} \right)\]
\[ \Rightarrow u = \tan^{- 1} \left( \tan\theta \right) . . . \left( i \right)\]
\[\text { And }\]
\[\text { Let, v } = \sin^{- 1} \left( 2x\sqrt{1 - x^2} \right)\]
\[ v = \sin^{- 1} \left( 2\sin\theta\sqrt{1 - \sin^2 \theta} \right)\]
\[ v = \sin^{- 1} \left( 2 \sin\theta\cos\theta \right)\]
\[ v = \sin^{- 1} \left( \sin2\theta \right) . . . \left( ii \right)\]
\[\text { Here,} \]
\[ - \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\]
\[ \Rightarrow - \frac{1}{\sqrt{2}} < \sin\theta < \frac{1}{\sqrt{2}}\]
\[ \Rightarrow - \frac{\pi}{4} < \theta < \frac{\pi}{4}\]
\[\text { So, from equation } \left( i \right), \]
\[u = \theta \left[ \text { Since,} \tan^{- 1} \left( \tan\theta \right) = \theta, \text { if } \theta \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \right]\]
\[ \Rightarrow u = \sin^{- 1} x\]
Differentiating it with respect to x,
\[\frac{du}{dx} = \frac{1}{\sqrt{1 - x^2}} . . . \left( iii \right)\]
\[\text{ from equation } \left( ii \right), \]
\[v = 2\theta \left[ \text { Since}, \sin^{- 1} \left( \sin\theta \right) = \theta, \text { if } \theta \in \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \right]\]
\[ \Rightarrow v = 2 \sin^{- 1} x\]
Differentiating it with respect to x,
\[\frac{dv}{dx} = \frac{2}{\sqrt{1 - x^2}} . . . \left( iv \right)\]
\[\text { Dividing equation } \left( iii \right) \text {by}\left( iv \right), \]
\[\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \left( \frac{1}{\sqrt{1 - x^2}} \right)\left( \frac{\sqrt{1 - x^2}}{2} \right)\]
\[ \therefore \frac{du}{dv} = \frac{1}{2}\]
संबंधित प्रश्न
If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Differentiate sin (log x) ?
Differentiate \[e^{\sin} \sqrt{x}\] ?
Differentiate \[3^{e^x}\] ?
Differentiate \[\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)\] ?
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?
Differentiate \[\frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}}\] ?
Differentiate \[e^{ax} \sec x \tan 2x\] ?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
If \[y = \sqrt{x + 1} + \sqrt{x - 1}\] , prove that \[\sqrt{x^2 - 1}\frac{dy}{dx} = \frac{1}{2}y\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^{2/3} + y^{2/3} = a^{2/3}\] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?
If \[xy \log \left( x + y \right) = 1\] ,Prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
Differentiate \[x^{1/x}\] with respect to x.
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[{10}^{ \log \sin x }\] ?
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?
If \[x^y \cdot y^x = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( y + x \log y \right)}{x \left( y \log x + x \right)}\] ?
If \[y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sec^2 x}{2 y - 1}\] ?
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 \[y = x \left| x \right|\] , find \[\frac{dy}{dx} \text{ for } x < 0\] ?
If \[- \frac{\pi}{2} < x < 0 \text{ and y } = \tan^{- 1} \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}, \text{ find } \frac{dy}{dx}\] ?
If f (x) is an odd function, then write whether `f' (x)` is even or odd ?
If \[y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =\] __________ .
If \[\sin y = x \sin \left( a + y \right), \text { then }\frac{dy}{dx} \text { is}\] ____________ .
If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .
Find the second order derivatives of the following function x3 log x ?
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
Let f(x) be a polynomial. Then, the second order derivative of f(ex) is
If x = 2 at, y = at2, where a is a constant, then \[\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}\] is
If y = etan x, then (cos2 x)y2 =
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
