I'm working through some partial differentiation examples in my study guide. Please see the work and let me know if it appears to be correct:
Suppose $$f(x,y) = e^{xy}$$ $$x(u,v) = 3u\sin{v}$$ $$y(u,v) = 4v^2u$$
For $g(u,v) = f(x(u,v), y(u,v))$, find the partial derivatives $\frac{\partial g}{\partial u}$ $\frac{\partial g}{\partial v}$:
$$g(u,v) = f(x(u,v),y(u,v))$$
$$\frac{\partial g}{\partial u} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial g}{\partial y}\frac{\partial y}{\partial u}$$
$$\frac{\partial g}{\partial v} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial g}{\partial y}\frac{\partial y}{\partial v}$$
$$\frac{\partial f}{\partial x} = e^{xy} \cdot y = ye^{xy}$$ $$\frac{\partial f}{\partial y} = e^{xy} \cdot x = xe^{xy}$$ $$\frac{\partial x}{\partial u} = 3\sin{v}$$ $$\frac{\partial y}{\partial u} = 4v^2$$ $$\frac{\partial x}{\partial v} = 3u(\cos{v} \cdot 1) = 3u\cos{v}$$ $$\frac{\partial y}{\partial v} = 8vu$$
Therefore
$$\frac{\partial g}{\partial u} = ye^{xy}(3\sin{v}) + xe^{xy}(4v^2)$$ $$= (4v^2u)e^{(3u\sin{v})(4v^2u)}(3\sin{v}) + (3u\sin{v})e^{(3u\sin{v})(4v^2u)}(4v^2)$$ $$= 12uv^2e^{(12u^2v^2\sin{v})}\sin{v} + 12uv^2e^{(12u^2v^2\sin{v})}\sin{v}$$ $$= 24uv^2e^{(12u^2v^2\sin{v})}\sin{v}$$
and
$$\frac{\partial g}{\partial v} = ye^{xy}(3u\cos{v}) + xe^{xy}(8uv)$$ $$= (4v^2u)e^{(3u\sin{v})(4v^2u)}(3u\cos{v}) + (3u\sin{v})e^{(3u\sin{v})(4v^2u)}(8uv)$$ $$= 12u^2v^2e^{(12u^2v^2\sin{v})}\cos{v} + 24u^2ve^{(12u^2v^2\sin{v})}\sin{v}$$
Thanks!