Question

find dy/dx of xsiny+cos2y=cosy

Answer 1

Use implicit differentiation.

Differentiate both sides of the equation with respect to x, treating y as a function of x.

x cosy dy/dx + sin y - 2 sin 2y dy/dx = -sin y dy/dx

Now solve for dy/dx. It will be a function of both x and y.

You could also differentiate both sides with respect to y, solve for dx/dy, and take the reciprocal of the answer. The result will look different but will still be correct.

x cos y + sin y dx/dy - 2 sin 2y = -sin y

dx/dy = [-siny + 2 sin2y -x cos y]/sin y
dy/dx = sin y/[-siny + 2 sin2y -x cos y]