Question

If ${\sin }^{2}y+\cos \left(xy\right)=\pi$, find $\frac{dy}{dx}$.

Answer 1

Consider the given equation.

${\sin }^{2}y+\cos xy=\pi$

Differentiate both sides with respect to $x$.

$2\sin y\times \cos y\times \frac{dy}{dx}-\sin xy\times (y+x\frac{dy}{dx})=0$

$2\sin y\cos y\frac{dy}{dx}-y\sin \left(xy\right)-x\sin \left(xy\right)\frac{dy}{dx}=0$

$2\sin y\cos y\frac{dy}{dx}-x\sin \left(xy\right)\frac{dy}{dx}=y\sin \left(xy\right)$

$(\sin 2y-x\sin \left(xy\right))\frac{dy}{dx}=y\sin \left(xy\right)$

$\frac{dy}{dx}=\frac{y\sin \left(xy\right)}{\sin 2y-x\sin \left(xy\right)}$

Hence, this is the required result.