# How do you implicitly differentiate -1=xytan(x/y) ?

Dec 27, 2015

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y}{x} \cdot \frac{2 x \csc \left(\frac{2 x}{y}\right) + y}{x \cot \left(\frac{x}{y}\right) - y}$

#### Explanation:

$- 1 = x y \tan \left(\frac{x}{y}\right)$

Differentiate both sides w.r.t. $x$.

$\frac{d}{\mathrm{dx}} \left(- 1\right) = \frac{d}{\mathrm{dx}} \left(x y \tan \left(\frac{x}{y}\right)\right)$

$0 = \frac{d}{\mathrm{dx}} \left(x y \tan \left(\frac{x}{y}\right)\right)$

Use the Product Rule, Chain Rule, and Quotient Rule.

$\frac{d}{\mathrm{dx}} \left(x y \tan \left(\frac{x}{y}\right)\right) = x y \frac{d}{\mathrm{dx}} \left(\tan \left(\frac{x}{y}\right)\right) + \tan \left(\frac{x}{y}\right) \frac{d}{\mathrm{dx}} \left(x y\right)$

$= x y \frac{d}{\mathrm{dx}} \left(\tan \left(\frac{x}{y}\right)\right) + \tan \left(\frac{x}{y}\right) \left(y \frac{d}{\mathrm{dx}} \left(x\right) + x \frac{d}{\mathrm{dx}} \left(y\right)\right)$

$= x y {\sec}^{2} \left(\frac{x}{y}\right) \frac{d}{\mathrm{dx}} \left(\frac{x}{y}\right) + \tan \left(\frac{x}{y}\right) \left(y + x \frac{\mathrm{dy}}{\mathrm{dx}}\right)$

$= x y {\sec}^{2} \left(\frac{x}{y}\right) \frac{y \frac{d}{\mathrm{dx}} \left(x\right) - x \frac{d}{\mathrm{dx}} \left(y\right)}{{y}^{2}} + \tan \left(\frac{x}{y}\right) \left(y + x \frac{\mathrm{dy}}{\mathrm{dx}}\right)$

$= \frac{x}{y} {\sec}^{2} \left(\frac{x}{y}\right) \left(y - x \frac{\mathrm{dy}}{\mathrm{dx}}\right) + \tan \left(\frac{x}{y}\right) \left(y + x \frac{\mathrm{dy}}{\mathrm{dx}}\right)$

$= x {\sec}^{2} \left(\frac{x}{y}\right) - \frac{{x}^{2}}{y} \frac{\mathrm{dy}}{\mathrm{dx}} + y \tan \left(\frac{x}{y}\right) + x \tan \left(\frac{x}{y}\right) \frac{\mathrm{dy}}{\mathrm{dx}}$

$= x {\sec}^{2} \left(\frac{x}{y}\right) + y \tan \left(\frac{x}{y}\right) + \left(x \tan \left(\frac{x}{y}\right) - \frac{{x}^{2}}{y}\right) \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{x {\sec}^{2} \left(\frac{x}{y}\right) + y \tan \left(\frac{x}{y}\right)}{x \tan \left(\frac{x}{y}\right) - \frac{{x}^{2}}{y}}$

$= \frac{y}{x} \cdot \frac{2 x \csc \left(\frac{2 x}{y}\right) + y}{x \cot \left(\frac{x}{y}\right) - y}$