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

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  • A function $f(x,y)$ of two variables $x$ and $y$,is said to be implicit,which are jumbled in such a way,that it is not possible to write y exclusively as a function of $x$.
  • $\large\frac{d}{dx}$$\phi(y)=\large\frac{d}{dy}.$$\phi(y).\large\frac{dy}{dx}$
$\sin(xy)+\large\frac{x}{y}$$=x^2-y$
Differentiating the above function w.r.t $x$ we get
(Apply product rule to differentiate $xy$ and quotient rule to differentiate $\large\frac{x}{y})$
$\cos(xy).[x\large\frac{dy}{dx}+$$y.1]+\large\frac{y.1-x dy/dx}{y^2}=$$2x-\large\frac{dy}{dx}$
$\Rightarrow \large\frac{dy}{dx}$$[x.y^2\cos(xy)-x-y^2]=2xy^2-y^3\cos(xy)-y$
$\large\frac{dy}{dx}=\large\frac{2xy^2-y-y^3\cos(xy)}{xy^2\cos(xy)-x-y^2}$
answered Jul 2, 2013 by sreemathi.v
 

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