# What is the implicit derivative of 25=cosy/x-3xy?

Mar 17, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{25 + 6 x y}{\sin y + 3 {x}^{2}}$

#### Explanation:

Differentiate both sides of the equation with respect to $x$:

$\frac{d}{\mathrm{dx}} \left(\cos \frac{y}{x} - 3 x y\right) = 0$

$\frac{- x \sin y \frac{\mathrm{dy}}{\mathrm{dx}} - \cos y}{x} ^ 2 - 3 \left(y + x \frac{\mathrm{dy}}{\mathrm{dx}}\right) = 0$

Since $x \ne 0$ we can multiply both sides by $x$:

$- \sin y \frac{\mathrm{dy}}{\mathrm{dx}} - \cos \frac{y}{x} - 3 x y - 3 {x}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$\frac{\mathrm{dy}}{\mathrm{dx}} \left(\sin y + 3 {x}^{2}\right) = - \cos \frac{y}{x} - 3 x y$

From the original expression we can substitute:

$\cos \frac{y}{x} = 25 + 3 x y$

so:

$\frac{\mathrm{dy}}{\mathrm{dx}} \left(\sin y + 3 {x}^{2}\right) = - 25 - 6 x y$

and finally:

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{25 + 6 x y}{\sin y + 3 {x}^{2}}$