I have $f(x,y) = \sin(x^2+y^2)$ where $y(x)=\sqrt{3\pi - x^2}$ and I am trying to find $f_x=\frac {\partial f}{\partial x}(x_0,y(x_0))$
What I tried doing is: $$ \frac {\partial f}{\partial x}= \frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}\cdot \frac {\partial y}{\partial x}$$ which gives me $$2x\cdot \cos(3\pi)+2\cdot\sqrt{3\pi-x^2}\cdot \cos(3\pi)\cdot \frac {-x}{\sqrt{3\pi-x^2}}$$
and that's equal to $0$.. Where am I wrong?
Hopefully you got $0$!
With your hypothesis, $f(x, y(x))$ is constant. Let us look at it more carefully:
$$f(x, y(x)) = sin\left(x^2 + y(x)^2\right) = sin\left(x^2 + (3\pi - x^2)\right) = sin(3\pi) = 0$$
That is enough to have $\dfrac{df\left(x, y(x)\right)}{dx} = 0$