Advertisements
Advertisements
प्रश्न
If \[f\left( x \right) = \frac{\sin^{- 1} x}{\sqrt{1 - x^2}}\] then (1 − x)2 f '' (x) − xf(x) =
विकल्प
1
−1
0
none of these
उत्तर
(a) 1
Here,
\[f\left( x \right) = \frac{\sin^{- 1} x}{\sqrt{1 - x^2}}\]
\[ \Rightarrow \sqrt{1 - x^2} f\left( x \right) = \sin^{- 1} x\]
\[\text { Diffferentiating w . r . t . x, we get }\]
\[\sqrt{1 - x^2} f^{'} \left( x \right) - \frac{x f\left( x \right)}{\sqrt{1 - x^2}} = \frac{1}{\sqrt{1 - x^2}}\]
\[ \Rightarrow \left( 1 - x^2 \right) f^{'} \left( x \right) - xf\left( x \right) = 1\]
DISCLAIMER : In the question instead of (1 − x)2 f '' (x) − xf(x)
it should be (1 − x)2 f ' (x) − xf(x) .
APPEARS IN
संबंधित प्रश्न
If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Differentiate the following function from first principles \[e^\sqrt{\cot x}\] .
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate (log sin x)2 ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{5 x}{1 - 6 x^2} \right), - \frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}}\] ?
If \[y = x \sin \left( a + y \right)\] ,Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = e^{3x} \sin 4x \cdot 2^x\] ?
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 \[x^y + y^x = \left( x + y \right)^{x + y} , \text{ find } \frac{dy}{dx}\] ?
If \[xy \log \left( x + y \right) = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?
If \[y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + ... to \infty}}}\], prove that \[\left( 2 y - 1 \right) \frac{dy}{dx} = \frac{1}{x}\] ?
Find \[\frac{dy}{dx}\], when \[x = a t^2 \text{ and } y = 2\ at \] ?
Find \[\frac{dy}{dx}\] , when \[x = \cos^{- 1} \frac{1}{\sqrt{1 + t^2}} \text{ and y } = \sin^{- 1} \frac{t}{\sqrt{1 + t^2}}, t \in R\] ?
If \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?
If \[x = a \left( \theta - \sin \theta \right) and, y = a \left( 1 + \cos \theta \right), \text { find } \frac{dy}{dx} \text{ at }\theta = \frac{\pi}{3} \] ?
If \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?
Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
Differentiate \[\sin^{- 1} \left( 2 ax \sqrt{1 - a^2 x^2} \right)\] with respect to \[\sqrt{1 - a^2 x^2}, \text{ if }-\frac{1}{\sqrt{2}} < ax < \frac{1}{\sqrt{2}}\] ?
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] _____________ .
For the curve \[\sqrt{x} + \sqrt{y} = 1, \frac{dy}{dx}\text { at } \left( 1/4, 1/4 \right)\text { is }\] _____________ .
Let \[\cup = \sin^{- 1} \left( \frac{2 x}{1 + x^2} \right) \text { and }V = \tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text { then } \frac{d \cup}{dV} =\] ____________ .
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 \[\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}\] is equal to ______________ .
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 log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0 ?
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find \[\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}\] ?
\[\text{ If x } = a\left( \cos t + \log \tan\frac{t}{2} \right) \text { and y } = a\left( \sin t \right), \text { evaluate } \frac{d^2 y}{d x^2} \text { at t } = \frac{\pi}{3} \] ?
If \[x = 3 \cos t - 2 \cos^3 t, y = 3\sin t - 2 \sin^3 t,\] find \[\frac{d^2 y}{d x^2} \] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
f(x) = xx has a stationary point at ______.
