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# If ${\mathrm{x}}^{2}-{\mathrm{y}}^{2}+3\mathrm{x}=5\mathrm{y}$ then $\frac{\mathrm{dy}}{\mathrm{dx}}=$

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a
$\frac{\sqrt{\mathrm{y}}\left(4\mathrm{x}\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}}\right)}{\sqrt{\mathrm{x}}\left(2\sqrt{\mathrm{y}}+\sqrt{\mathrm{x}}\right)}$
b
$\frac{\sqrt{\mathrm{y}}\left(1-2\sqrt{\mathrm{xy}}-\mathrm{y}\right)}{\sqrt{\mathrm{x}}\left(1+2\sqrt{\mathrm{xy}}+\mathrm{x}\right)}$
c
$\frac{2\mathrm{x}+3}{2\mathrm{y}+5}$
d
$\frac{\sqrt{1-{y}^{2}}-\sqrt{\left(1-{x}^{2}\right)\left(1-{y}^{2}\right)}}{\sqrt{1-{x}^{2}}-\sqrt{\left(1-{x}^{2}\right)\left(1-{y}^{2}\right)}}$
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detailed solution

Correct option is C

The given equation is ${\mathrm{x}}^{2}-{\mathrm{y}}^{2}+3\mathrm{x}=5\mathrm{y}$

Use implicit differentiation, differentiate both sides with respect to $x$

$\begin{array}{rcl}2x-2y\frac{dy}{dx}+3& =& 5\frac{dy}{dx}\\ \frac{dy}{dx}\left(2y+5\right)& =& 2x+3\\ \frac{dy}{dx}& =& \overline{)\frac{\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{3}}{\mathbf{2}\mathbf{y}\mathbf{+}\mathbf{5}}}\end{array}$

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