Advertisements
Advertisements
Question
If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
Advertisements
Solution
We have,
`xy = 1`
Differentiating with respect to x, we get,
\[\frac{d}{dx}\left( xy \right) = \frac{d}{dx}\left( 1 \right)\]
\[ \Rightarrow x\frac{d y}{d x} + y\frac{d}{dx}\left( x \right) = 0 ..........\left[ \text{ Using product rule} \right]\]
\[ \Rightarrow x\frac{d y}{d x} + y\left( 1 \right) = 0\]
\[ \Rightarrow \frac{d y}{d x} = - \frac{y}{x} \]
\[ \Rightarrow \frac{d y}{d x} = - \frac{y}{\frac{1}{y}} ..........\left[ \because x = \frac{1}{y} \right]\]
\[ \Rightarrow \frac{d y}{d x} = - y^2 \]
\[ \Rightarrow \frac{d y}{d x} + y^2 = 0\]
APPEARS IN
RELATED QUESTIONS
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `cos^(-1)(1/sqrt3)`
Differentiate the following function from first principles \[e^\sqrt{\cot x}\] .
Differentiate sin (log x) ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[e^{\tan 3 x} \] ?
Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?
Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\]?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
If \[y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}\] prove that \[\frac{dy}{dx} = \sec 2x\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sin x + \cos x}{\sqrt{2}} \right\}, - \frac{3 \pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?
If \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, - \frac{1}{2} < x < 0, \text{ find } \frac{dy}{dx} \] ?
Find \[\frac{dy}{dx}\] in the following case: \[y^3 - 3x y^2 = x^3 + 3 x^2 y\] ?
If \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?
Differentiate \[x^{\tan^{- 1} x }\] ?
Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)\] ?
If \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?
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 \[x = b \sin^2 \theta \text{ and } y = a \cos^2 \theta\] ?
If \[x = a\left( t + \frac{1}{t} \right) \text{ and y } = a\left( t - \frac{1}{t} \right)\] ,prove that \[\frac{dy}{dx} = \frac{x}{y}\]?
If \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?
If \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?
Differentiate \[\left( \cos x \right)^{\sin x }\] with respect to \[\left( \sin x \right)^{\cos x }\]?
Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
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 \[f\left( 1 \right) = 4, f'\left( 1 \right) = 2\] find the value of the derivative of \[\log \left( f\left( e^x \right) \right)\] w.r. to x at the point x = 0 ?
If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text{ find } \frac{dy}{dx}\] ?
If f (x) is an odd function, then write whether `f' (x)` is even or odd ?
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
If y = 3 cos (log x) + 4 sin (log x), prove that x2y2 + xy1 + y = 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 \[\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}\] ?
\[\text { If x } = a\left( \cos2t + 2t \sin2t \right)\text { and y } = a\left( \sin2t - 2t \cos2t \right), \text { then find } \frac{d^2 y}{d x^2} \] ?
\[\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3} \] ?
If x = t2, y = t3, then \[\frac{d^2 y}{d x^2} =\]
