Advertisements
Advertisements
प्रश्न
If \[y = \sqrt{x^2 + a^2}\] prove that \[y\frac{dy}{dx} - x = 0\] ?
Advertisements
उत्तर
\[\text{ We have, y } = \sqrt{x^2 + a^2}\]
Squaring both sides we get,
\[y^2 = x^2 + a^2 \]
\[ \Rightarrow 2y\frac{d y}{d x} = \frac{d}{dx}\left( x^2 + a^2 \right)\]
\[ \Rightarrow 2y\frac{d y}{d x} = \left( 2x \right)\]
\[ \Rightarrow 2y\frac{d y}{d x} = 2x\]
\[ \Rightarrow y\frac{d y}{d x} = x\]
\[ \Rightarrow y\frac{d y}{d x} - x = 0\]
\[\text{ Hence proved }\]
APPEARS IN
संबंधित प्रश्न
Differentiate \[\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[e^x \log \sin 2x\] ?
If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?
If \[y = e^x + e^{- x}\] prove that \[\frac{dy}{dx} = \sqrt{y^2 - 4}\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{1 + \sqrt{1 - x^2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{2^{x + 1}}{1 - 4^x} \right), - \infty < x < 0\] ?
Differentiate the following with respect to x:
\[\cos^{- 1} \left( \sin x \right)\]
Find \[\frac{dy}{dx}\] in the following case \[\left( x^2 + y^2 \right)^2 = xy\] ?
Find \[\frac{dy}{dx}\] in the following case \[\sin xy + \cos \left( x + y \right) = 1\] ?
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 \[y = x \sin y\] , Prove that \[\frac{dy}{dx} = \frac{\sin y}{\left( 1 - x \cos y \right)}\] ?
Differentiate \[x^{1/x}\] with respect to x.
Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \sin x \right)^{\cos x} + \left( \cos x \right)^{\sin x}\] ?
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
If \[e^y = y^x ,\] prove that\[\frac{dy}{dx} = \frac{\left( \log y \right)^2}{\log y - 1}\] ?
If \[y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + . . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sin x}{1 - 2 y}\] ?
Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?
Find \[\frac{dy}{dx}\] when \[x = \frac{2 t}{1 + t^2} \text{ and } y = \frac{1 - t^2}{1 + t^2}\] ?
If \[x = a \left( \frac{1 + t^2}{1 - t^2} \right) \text { and y } = \frac{2t}{1 - t^2}, \text { find } \frac{dy}{dx}\] ?
Differentiate \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\] \[x \in \left( - 1, 0 \right)\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?
Differentiate \[\cos^{- 1} \left( 4 x^3 - 3x \right)\] with respect to \[\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), \text{ if }\frac{1}{2} < x < 1\] ?
If \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?
If \[\frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \text { and y } = \sin^{- 1} \left( \sin x \right), \text { find } \frac{dy}{dx} \] ?
If \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?
If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
Given \[f\left( x \right) = 4 x^8 , \text { then }\] _________________ .
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 \[y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =\] __________ .
Find the second order derivatives of the following function tan−1 x ?
If y = x3 log x, prove that \[\frac{d^4 y}{d x^4} = \frac{6}{x}\] ?
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then\[\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 =\]
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) w . r . t . \sin^{- 1} \frac{2x}{1 + x^2},\]tan-11+x2-1x w.r.t. sin-12x1+x2, if x ∈ (–1, 1) .
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
f(x) = xx has a stationary point at ______.
