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Find $\large \frac{dy}{dx}$ when x and y are connected by the relation $(x^2+y^2)^2=xy$

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Toolbox:
  • A function $f(x,y)$ is said to be implicit if it is jumbled in such a way,that it is not possible to write $y$ exclusively s a function of $x$.
  • $\large\frac{d}{dx}$$\phi(y)=\large\frac{d}{dy}$$\phi(y).\large\frac{dy}{dx}$
Step 1:
$(x^2+y^2)^2=xy$
Differentiating w.r.t $x$ we get,
Apply chain rule to differentiate $(x^2+y^2)^2$ and product rule to differentiate $xy$
$2(x^2+y^2).(2x+2y\large\frac{dy}{dx})$$=x.\large\frac{dy}{dx}$$+y.1$
Step 2:
$\Rightarrow 2(x^2+y^2).2(x+y\large\frac{dy}{dx})$$=x.\large\frac{dy}{dx}$$+y$
$\Rightarrow \large\frac{dy}{dx}$$[4(x^2+y^2).y-x]=y-4x(x^2+y^2)$
$\Rightarrow \large\frac{dy}{dx}=\large\frac{y-4x(x^2+y^2)}{4y(x^2+y^2)-x}$
answered Jul 2, 2013 by sreemathi.v
 

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