# How do you implicitly differentiate x^2 y^3 − xy = 10?

Mar 8, 2016

You can do it like this

#### Explanation:

${x}^{2} {y}^{3} - x y = 10$

Differentiate both sides with respect to $x$:

$D \left({x}^{2} {y}^{3} - x y\right) = D \left(10\right)$

Then apply the product rule and the chain rule:

${x}^{2} 3 {y}^{2} y ' + {y}^{3} 2 x - \left(x y ' + y\right) = 0$

$3 {x}^{2} {y}^{2} y ' + 2 {y}^{3} x - x y ' - y = 0$

$\therefore y ' \left(3 {x}^{2} {y}^{2} - x\right) = y - 2 {y}^{3} x$

$\therefore y ' \left(3 {x}^{2} {y}^{2} - x\right) = y \left(1 - 2 x {y}^{2}\right)$

$\therefore y ' = \frac{y \left(1 - 2 x {y}^{2}\right)}{\left(3 {x}^{2} {y}^{2} - x\right)}$

y'=(y(1-2xy^2))/(x(3xy^2-1)