How do you differentiate xy=cot(xy)?

Jul 23, 2015

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{y}{x}$

Explanation:

I am assuming that you want to find $\frac{\mathrm{dy}}{\mathrm{dx}}$.

We shall use both implicit differentiation and chain rule.
$x y = \cot \left(x y\right)$

Differentiate both sides with respect to $x$ to get:
$\frac{d}{\mathrm{dx}} \left(x y\right) = \frac{d}{\mathrm{dx}} \left(\cot \left(x y\right)\right)$

Apply chain rule
$\frac{d}{\mathrm{dx}} \left(x y\right) = - {\csc}^{2} \left(x y\right) \frac{d}{\mathrm{dx}} \left(x y\right)$
$\implies \left(1 + {\csc}^{2} \left(x y\right)\right) \frac{d}{\mathrm{dx}} \left(x y\right) = 0$
$\implies \frac{d}{\mathrm{dx}} \left(x y\right) = 0$ (Since $\left(1 + {\csc}^{2} \left(x y\right)\right)$ is always positive)

Apply chain rule again
$y + x \frac{\mathrm{dy}}{\mathrm{dx}} = 0$
$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{y}{x}$