Advertisements
Advertisements
प्रश्न
Find \[\frac{dy}{dx}\] in the following case \[x^5 + y^5 = 5 xy\] ?
उत्तर
\[\text{We have, }x^5 + y^5 = 5xy\]
Differentiating with respect to x, we get,
\[\frac{d}{dx}\left( x^5 \right) + \frac{d}{dx}\left( y^5 \right) = \frac{d}{dx}\left( 5xy \right)\]
\[ \Rightarrow 5 x^4 + 5 y^4 \frac{d y}{d x} = 5\left[ x\frac{d y}{d x} + y\frac{d}{d x}\left( x \right) \right]\]
\[ \Rightarrow 5 x^4 + 5 y^4 \frac{d y}{d x} = 5\left[ x\frac{d y}{d x} + y\left( 1 \right) \right]\]
\[ \Rightarrow 5 x^4 + 5 y^4 \frac{d y}{d x} = 5x\frac{d y}{d x} + 5y\]
\[ \Rightarrow 5 y^4 \frac{d y}{d x} - 5x\frac{d y}{d x} = 5y - 5 x^4 \]
\[ \Rightarrow 5\frac{d y}{d x}\left( y^4 - x \right) = 5\left( y - x^4 \right)\]
\[ \Rightarrow \frac{d y}{d x} = \frac{5\left( y - x^4 \right)}{5\left( y^4 - x \right)}\]
\[ \Rightarrow \frac{d y}{d x} = \frac{y - x^4}{y^4 - x}\]
APPEARS IN
संबंधित प्रश्न
Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?
Differentiate \[e^\sqrt{\cot x}\] ?
Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?
Differentiate \[\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)\] ?
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
Differentiate \[\cos \left( \log x \right)^2\] ?
If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?
Differentiate \[\cos^{- 1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right), < x < \infty\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), 0 < x < 1,\] prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
If \[\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)\] Prove that \[\frac{dy}{dx} = \frac{x + y}{x - y}\] ?
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 \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?
If \[\sqrt{y + x} + \sqrt{y - x} = c, \text {show that } \frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
If \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?
If \[y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + ... to \infty}}}\], prove that \[\left( 2 y - 1 \right) \frac{dy}{dx} = \frac{1}{x}\] ?
If \[\frac{dy}{dx}\] when \[x = a \cos \theta \text{ and } y = b \sin \theta\] ?
Differentiate x2 with respect to x3
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( - \frac{1}{2}, - \frac{1}{2 \sqrt{2}} \right)\] ?
If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
If \[f\left( x \right) = \log \left\{ \frac{u \left( x \right)}{v \left( x \right)} \right\}, u \left( 1 \right) = v \left( 1 \right) \text{ and }u' \left( 1 \right) = v' \left( 1 \right) = 2\] , then find the value of `f' (1)` ?
If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .
If \[\sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \text { log a then } \frac{dy}{dx}\] is equal to _____________ .
If \[y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] __________ .
Find the second order derivatives of the following function sin (log x) ?
If y = e−x cos x, show that \[\frac{d^2 y}{d x^2} = 2 e^{- x} \sin x\] ?
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 = 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 y = cosec−1 x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?
If y = a xn + 1 + bx−n and \[x^2 \frac{d^2 y}{d x^2} = \lambda y\] then write the value of λ ?
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 y = etan x, then (cos2 x)y2 =
