# How do you implicitly differentiate  3x^2 - 2-xy + y/(2x) = 11y ?

Dec 5, 2015

Apply implicit differentiation to find
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{12 {x}^{3} - y \left(2 {x}^{2} + 1\right)}{x \left(2 {x}^{2} + 22 {x}^{2} - 1\right)}$

#### Explanation:

Using Implicit Differentiation requires use of the chain rule in general. In this case we will also use the product rule and the quotient rule.

$\frac{d}{\mathrm{dx}} \left(3 {x}^{2} - 2 - x y + \frac{y}{2 x}\right) = \frac{d}{\mathrm{dx}} \left(11 y\right)$

$\implies \frac{d}{\mathrm{dx}} 3 {x}^{2} - \frac{d}{\mathrm{dx}} 2 - \frac{d}{\mathrm{dx}} x y + \frac{d}{\mathrm{dx}} \frac{y}{2 x} = \frac{d}{\mathrm{dx}} 11 y$

$\implies 6 x - 0 - \left(1 \cdot y + x \frac{\mathrm{dy}}{\mathrm{dx}}\right) + \frac{2 x \frac{\mathrm{dy}}{\mathrm{dx}} - 2 y}{4 {x}^{2}} = 11 \frac{\mathrm{dy}}{\mathrm{dx}}$

$\implies 6 x - y - x \frac{\mathrm{dy}}{\mathrm{dx}} + \frac{1}{2 x} \frac{\mathrm{dy}}{\mathrm{dx}} - \frac{y}{2 {x}^{2}} = 11 \frac{\mathrm{dy}}{\mathrm{dx}}$

$\implies 11 \frac{\mathrm{dy}}{\mathrm{dx}} + x \frac{\mathrm{dy}}{\mathrm{dx}} - \frac{1}{2 x} \frac{\mathrm{dy}}{\mathrm{dx}} = 6 x - y - \frac{y}{2 {x}^{2}}$

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} \left(11 + x - \frac{1}{2 x}\right) = 6 x - y - \frac{y}{2 {x}^{2}}$

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6 x - y - \frac{y}{2 {x}^{2}}}{11 + x - \frac{1}{2 x}}$

$= \frac{12 {x}^{3} - y \left(2 {x}^{2} + 1\right)}{x \left(2 {x}^{2} + 22 {x}^{2} - 1\right)}$