Advertisements
Advertisements
Question
If \[xy \log \left( x + y \right) = 1\] ,Prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
Advertisements
Solution
\[\text{ We have, xy } \log\left( x + y \right) = 1\]
Differentiating it with respect to x,
\[\Rightarrow \frac{d}{dx}\left[ xy \log\left( x + y \right) \right] = \frac{d}{dx}\left( 1 \right)\]
\[ \Rightarrow xy\frac{d}{dx}\log\left( x + y \right) + x \log\left( x + y \right)\frac{d y}{d x} + y \log\left( x + y \right)\frac{d}{dx}\left( x \right) = 0 \left[ \text{ using chain rule and product rule } \right]\]
\[ \Rightarrow xy\left( \frac{1}{x + y} \right)\frac{d}{dx}\left( x + y \right) + x \log\left( x + y \right)\frac{d y}{d x} + y \log\left( x + y \right)\left( 1 \right) = 0\]
\[ \Rightarrow \left( \frac{xy}{x + y} \right) \left( 1 + \frac{d y}{d x} \right) + x \log\left( x + y \right)\frac{d y}{d x} + y \log\left( x + y \right) = 0\]
\[ \Rightarrow \left( \frac{xy}{x + y} \right)\frac{d y}{d x} + \left( \frac{xy}{x + y} \right) + x\left( \frac{1}{xy} \right)\frac{d y}{d x} + y\left( \frac{1}{xy} \right) = 0 \left[ \because xy \log\left( x + y \right) = 1 \right]\]
\[ \Rightarrow \frac{d y}{d x}\left[ \frac{xy}{x + y} + \frac{1}{y} \right] = - \left[ \frac{1}{x} + \frac{xy}{x + y} \right]\]
\[ \Rightarrow \frac{d y}{d x}\left[ \frac{x y^2 + x + y}{\left( x + y \right)y} \right] = - \left[ \frac{x + y + x^2 y}{x\left( x + y \right)} \right]\]
\[ \Rightarrow \frac{d y}{d x} = - \left[ \frac{x + y + x^2 y}{x\left( x + y \right)} \right]\left[ \frac{y\left( x + y \right)}{x y^2 + x + y} \right]\]
\[ \Rightarrow \frac{d y}{d x} = - \frac{y}{x}\left( \frac{x + y + x^2 y}{x + y + x y^2} \right)\]
Hence proved
APPEARS IN
RELATED QUESTIONS
Differentiate tan2 x ?
Differentiate `2^(x^3)` ?
Differentiate \[3^{e^x}\] ?
Differentiate \[\tan \left( e^{\sin x }\right)\] ?
Differentiate \[e^{\sin^{- 1} 2x}\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
If \[y = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] .prove that \[\frac{dy}{dx} = 1 - y^2\] ?
If \[y = \left( x - 1 \right) \log \left( x - 1 \right) - \left( x + 1 \right) \log \left( x + 1 \right)\] , prove that \[\frac{dy}{dc} = \log \left( \frac{x - 1}{1 + x} \right)\] ?
If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?
Differentiate \[\sin^{- 1} \left( 2 x^2 - 1 \right), 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x \in R\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
If \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\] \[\frac{dy}{dx} \] ?
Find \[\frac{dy}{dx}\] in the following case \[x^5 + y^5 = 5 xy\] ?
Differentiate \[x^\left( \sin x - \cos x \right) + \frac{x^2 - 1}{x^2 + 1}\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
find \[\frac{dy}{dx}\] \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\sin x} + \left( \sin x \right)^x\] ?
If \[y = x \sin y\] , prove that \[\frac{dy}{dx} = \frac{y}{x \left( 1 - x \cos y \right)}\] ?
If \[\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}\] ?
\[y = \left( \sin x \right)^{\left( \sin x \right)^{\left( \sin x \right)^{. . . \infty}}} \],prove that \[\frac{y^2 \cot x}{\left( 1 - y \log \sin x \right)}\] ?
If \[y = e^{x^{e^x}} + x^{e^{e^x}} + e^{x^{x^e}}\], prove that \[\frac{dy}{dx} = e^{x^{e^x}} \cdot x^{e^x} \left\{ \frac{e^x}{x} + e^x \cdot \log x \right\}+ x^{e^{e^x}} \cdot e^{e^x} \left\{ \frac{1}{x} + e^x \cdot \log x \right\} + e^{x^{x^e}} x^{x^e} \cdot x^{e - 1} \left\{ x + e \log x \right\}\]
If \[x = 2 \cos \theta - \cos 2 \theta \text{ and y} = 2 \sin \theta - \sin 2 \theta\], prove that \[\frac{dy}{dx} = \tan \left( \frac{3 \theta}{2} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
If \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?
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 \[\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}\] is equal to ______________ .
If y = x + tan x, show that \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?
If x = a cos θ, y = b sin θ, show that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
If y = 3 cos (log x) + 4 sin (log x), prove that x2y2 + xy1 + y = 0 ?
If log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0 ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
If \[x = 3 \cos t - 2 \cos^3 t, y = 3\sin t - 2 \sin^3 t,\] find \[\frac{d^2 y}{d x^2} \] ?
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) .
