# How do you use implicit differentiation to find y' for sin(xy) = 1?

Sep 12, 2014

By implicit differentiation, we can find
$y ' = \frac{1}{x \cos \left(x y\right)} - \frac{y}{x}$.

Let us work through it.

$\sin \left(x y\right) = 1$

by implicitly differentiating with respect to $x$,
$R i g h t a r r o w \cos \left(x y\right) \cdot \left(1 \cdot y + x \cdot y '\right) = 1$

by dividing by $\cos \left(x y\right)$,
$R i g h t a r r o w y + x y ' = \frac{1}{\cos} \left(x y\right)$

by subtracting $y$,
$R i g h t a r r o w x y ' = \frac{1}{\cos} \left(x y\right) - y$

by dividing by $x$,
$R i g h t a r r o w y ' = \frac{1}{x \cos \left(x y\right)} - \frac{y}{x}$