Q: Suppose $\displaystyle f$ is a differentiable function of $\displaystyle x$ and $\displaystyle y$, and $\displaystyle g(u, v) = f(e^u + sin(v), e^u + cos(v))$. Use the table of values to calculate $\displaystyle g_{u}(0, 0)$ and $\displaystyle g_{v}(0, 0)$.

| $\displaystyle f$ | $\displaystyle g$ | $\displaystyle f_{x}$ | $\displaystyle f_{y}$ |

(0, 0) | 3 | 6 | 4 | 8 |

(1, 2) | 6 | 3 | 2 | 5 |

I apply the chain rule for 2 independent/2 dependent variable how I /think/ it's suppose to be applied in this problem:

$\displaystyle \frac{\partial x}{\partial u}=e^u$, $\displaystyle \frac{\partial x}{\partial v}=cos(v)$

$\displaystyle \frac{\partial y}{\partial u}=e^u$, $\displaystyle \frac{\partial y}{\partial v}=-sin(v)$

$\displaystyle \frac{\partial g}{\partial u}=\frac{\partial g}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial g}{\partial y}\frac{\partial y}{\partial u} = 4*e^u + 8*e^u = 12$