Question:
Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in each of the following:
$x y=c^{2}$
Solution:
We are given with an equation $x y=c^{2} ;$ we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get,
By using the product rule on the left hand side,
$\frac{\mathrm{d}(\mathrm{xy})}{\mathrm{dx}}=\frac{\mathrm{dc}^{2}}{\mathrm{dx}}$
$x \frac{d y}{d x}+y(1)=0$
$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\mathrm{y}}{\mathrm{x}}$
Or we can further solve it by putting the value of $y$,
$\frac{d y}{d x}=\frac{-c^{2}}{x^{2}}$