Advertisements
Advertisements
प्रश्न
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
उत्तर
Here,
\[y = \left( \cot^{- 1} x \right)^2 \]
\[\text { Differentiating w . r . t . x, we get }\]
\[ y_1 = 2co t^{- 1} x \times \frac{- 1}{1 + x^2} = \frac{- 2co t^{- 1} x}{1 + x^2}\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[ y_2 = \frac{2 + 4x \cot^{- 1} x}{\left( 1 + x^2 \right)^2}\]
\[ \Rightarrow y_2 = \frac{2}{\left( 1 + x^2 \right)^2} + \frac{2x \times 2 \cot^{- 1} x}{\left( 1 + x^2 \right)\left( 1 + x^2 \right)}\]
\[ \Rightarrow y_2 = \frac{2}{\left( 1 + x^2 \right)^2} - \frac{2x y_1}{\left( 1 + x^2 \right)}\]
\[ \Rightarrow \left( 1 + x^2 \right)^2 y_2 = 2 - 2x y_1 \left( 1 + x^2 \right)\]
\[ \Rightarrow \left( 1 + x^2 \right)^2 y_2 + 2x y_1 \left( 1 + x^2 \right) = 2\]
Hence proved.
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles e3x.
Differentiate the following functions from first principles ecos x.
Differentiate the following functions from first principles x2ex ?
Differentiate tan (x° + 45°) ?
Differentiate \[3^{x \log x}\] ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?
If \[x \sqrt{1 + y} + y \sqrt{1 + x} = 0\] , prove that \[\left( 1 + x \right)^2 \frac{dy}{dx} + 1 = 0\] ?
Differentiate \[x^{\tan^{- 1} x }\] ?
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 \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
If \[y = x \sin y\] , prove that \[\frac{dy}{dx} = \frac{y}{x \left( 1 - x \cos y \right)}\] ?
If \[y = \left( \cos x \right)^{\left( \cos x \right)^{\left( \cos x \right) . . . \infty}}\],prove that \[\frac{dy}{dx} = - \frac{y^2 \tan x}{\left( 1 - y \log \cos x \right)}\]?
If \[x = a\left( t + \frac{1}{t} \right) \text{ and y } = a\left( t - \frac{1}{t} \right)\] ,prove that \[\frac{dy}{dx} = \frac{x}{y}\]?
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( \frac{1}{2 \sqrt{2}}, \frac{1}{2} \right)\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right), \text { if }- \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
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 \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?
If \[\frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \text { and y } = \sin^{- 1} \left( \sin x \right), \text { find } \frac{dy}{dx} \] ?
Find the second order derivatives of the following function ex sin 5x ?
If y = x3 log x, prove that \[\frac{d^4 y}{d x^4} = \frac{6}{x}\] ?
If x = a (1 − cos3 θ), y = a sin3 θ, prove that \[\frac{d^2 y}{d x^2} = \frac{32}{27a} \text { at } \theta = \frac{\pi}{6}\] ?
If x = cos θ, y = sin3 θ, prove that \[y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 = 3 \sin^2 \theta\left( 5 \cos^2 \theta - 1 \right)\] ?
If \[y = e^{\tan^{- 1} x}\] prove that (1 + x2)y2 + (2x − 1)y1 = 0 ?
If \[y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2\] show that \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2\] ?
If y = ae2x + be−x, show that, \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\] ?
\[\text { If x } = \cos t + \log \tan\frac{t}{2}, y = \sin t, \text { then find the value of } \frac{d^2 y}{d t^2} \text { and } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
If x = 2at, y = at2, where a is a constant, then find \[\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}\] ?
If x = a cos nt − b sin nt, then \[\frac{d^2 x}{d t^2}\] is
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
If \[y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)\], then \[\frac{d^2 y}{d x^2} =\]
If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
If y = etan x, then (cos2 x)y2 =
Find the minimum value of (ax + by), where xy = c2.
