Differentiation through product of variables I am working on a problem connected to shallow water waves. 
I have a vector:
$U = \begin{bmatrix} h \\
h \cdot v_1\\
h \cdot v_2\end{bmatrix}$
and a function
$f(U) = \begin{bmatrix} h \cdot v_1 \\
h \cdot v_1^2 + 0.5\cdot gh^2\\
h \cdot v_1 \cdot v_2\end{bmatrix}$
I now want to calculate the Jacobian Matrix of $f(U)$.  
However, I am lost at how to calculate the partial derivatives when it comes to differentiating through a product. I.e.:
$\frac{\partial (h \cdot v_1^2 + 0.5 \cdot gh^2)}{\partial (h \cdot v_1)} = \frac{\partial (h \cdot v_1^2) }{\partial (h \cdot v_1)} + \frac{\partial (0.5 \cdot gh^2)}{\partial (h \cdot v_1)} = v_1 + ? \dots$
or    
$\frac{\partial (h \cdot v_1^2 + 0.5 \cdot gh^2)}{\partial (h \cdot v_2)} = \frac{\partial (h \cdot v_1^2) }{\partial (h \cdot v_2)} + \frac{\partial (0.5 \cdot gh^2)}{\partial (h \cdot v_2)} = \dots$
Googling it is really difficult and brought no result, since I only ever find explanations for the product rule...
Maybe someone here could enlighten me! Any kind of pointer in the right direction is highly appreciated! Thanks so much in advance!
 A: To do this, simply get rid of the "products of variables" by a substitution.
First, set $w_1 = h v_1$ and $w_2 = h v_2$.
Next, solve for $v_1 = w_1 / h$ and $v_2 = w_2 / h$.
Next, substitute for $v_1,v_2$ and simplify: 
$U = \begin{bmatrix} h \\
w_1\\
w_2\end{bmatrix}$
$f(U) = \begin{bmatrix} w_1 \\
w_1^2 \bigm/ h + 0.5\cdot gh^2\\
w_1w_2 \bigm/ h
\end{bmatrix}$
Next, compute the Jacobian Matrix for $f(U)$ in the usual way.
Finally, substitute for $w_1$ and $w_2$ and simplify.
Remarks: In your comment you ask about the "very first" entry of the Jacobian matrix.
The meaning of partial derivatives depends on the full set of coordinate variables, not just on a single variable. 
In your problem, you have informed us that the coordinate variables are $h$, $hv_1$, $hv_2$. I have simply subsituted these with one letter symbols $h$, $w_1$, $w_2$, respectively.
The meaning of the partial derivative $\frac{\partial w_1}{\partial h}$, using the coordinate variables you specified, means that you hold $w_1$ and $w_2$ constant and, while holding them constant, you vary $h$ and take the derivative of $w_1$ with respect to $h$. The result is zero, because $w_1$ has been held constant.
You can translate this back into your own notation: the meaning of $\frac{\partial hv_1}{\partial h}$ is that you hold $hv_1$ and $hv_2$ constant and, while holding them constant, you take the derivative of $hv_1$ with respect to $h$. Since $hv_1$ is held constant, its derivative is zero.
In other words, the definition of the partial derivative $\frac{\partial}{\partial h}$ is dependent on which other two coordinates you choose: its definition with coordinates $w_1,w_2$ is not the same as its definition with coordinates $v_1,v_2$. You can verify this for yourself if you look up the actual definition of partial derivatives, as a limit of difference quotients. See here for a discussion.
A: In the final analysis your $f$ is a function $f:\>{\mathbb R}^3\to{\mathbb R}^3$ taking the variables $h$, $v_1$, $v_2$ as input and producing three scalar values
$$u:=hv_1,\quad v:=hv_1^2+{g\over2}h^2,\quad w:=hv_1v_2$$
as output, whereby $g$ seems to be some constant. By definition the Jacobian of $f$ is the matrix of partial derivatives
$$\left[\matrix{u_h& u_{v_1}&u_{v_2}\cr v_h& v_{v_1}&v_{v_2}\cr
w_h& w_{v_1}&w_{v_2}\cr}\right]=\left[\matrix{v_1& h&0\cr v_1^2+gh& 2v_1h&0\cr
v_1v_2&hv_2&hv_1\cr}\right]\ .$$
