Question

E^y*cos x = 1 + sin(xy)

Answer 1

I assume you are studying implicit differentiation. Solving for dy/dx.
For the Left side use product rule. For Right side, use chain rule. Remember any time you take derivative of "y-term" you must multiply it by dy/dx.

---> $\frac{dy}{dx} e^y cos (x) - e^y sin(x) = (y + x \frac{dy}{dx}) cos (xy)$

Get dy/dx terms on one side:
$\frac{dy}{dx} e^y cos (x) - \frac{dy}{dx} x cos(xy) = e^y sin(x) + y cos (xy)$

Factor out dy/dx and solve:
$\frac{dy}{dx} (e^y cos (x) - x cos(xy)) = e^y sin(x) + y cos (xy) \\.......\\ \frac{dy}{dx} = \frac{e^y sin(x) + y cos (xy) }{e^y cos (x) - x cos(xy)}$