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Find \( \frac {dy}{dx} \) in the following: \( x^2 +xy + y^2 = 100 \)

$\begin{array}{1 1} -\large \frac{2x+y}{2y+x} \\ \large \frac{2x+y}{2y+x} \\ -\large \frac{2y+x}{2x+y} \\ \large \frac{2y+x}{2x+y} \end{array} $

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  • For equations that are not of the form $y = f(x)$, we need to differentiate each term separately on LHS and RHS and then calculate $\large \frac{dy}{dx}$
Given $x^2+xy+y^2=100$
This is not a standard differentiation of the form $y = f(x)$. We need to differentiate each term separately on LHS and RHS and then calculate $\large \frac{dy}{dx}$
According to the Product Rule for differentiation, given two functions $u$ and $v, \large \frac {d(uv)}{dx} $$= u \large \frac{dv}{dx}$$+ v \large \frac{du}{dx}$
Differentiating both sides:
$\Rightarrow 2x\;dx+x\;dy+y\;dx+2y\;dy = 0$
$\Rightarrow (2y + x)\; dy = -(2x+y)\; dx$
$\Rightarrow \large \frac{dy}{dx} $$= -\large \frac{2x+y}{2y+x}$
answered Apr 5, 2013 by balaji.thirumalai
 

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