Question: Let $f: U \to \mathbb{R}$ and $g: V \to \mathbb{R}$, where $U \subset \mathbb{R}^n$ and $V \subset \mathbb{R}^n$ both open. Now suppose we take the product of these two functions, i.e. formally the product rule

$$\mathbf{D}(fg)(a) = g\mathbf{D}f(a) + f\mathbf{D}g(a)$$

How would we construct a formula for the Jacobian of $f \cdot g$?

What I have so far is that

$$ \begin{align*} g\mathbf{D}f(a) + f\mathbf{D}g(a) &= g\mathbf{D} \begin{pmatrix} f_1(\mathbf{a}) \\ \vdots \\ f_n(\mathbf{a}) \end{pmatrix} + f\mathbf{D} \begin{pmatrix} g_1(\mathbf{a}) \\ \vdots \\ g_n(\mathbf{a}) \end{pmatrix} \\ &= g \begin{pmatrix} \frac{\partial f_1}{\partial a_1} & \dots & \frac{\partial f_1}{\partial a_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_n}{a_1} & \dots & \frac{\partial f_n}{\partial a_n} \end{pmatrix} + f \begin{pmatrix} \frac{\partial g_1}{\partial a_1} & \dots & \frac{\partial g_1}{\partial a_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial g_n}{a_1} & \dots & \frac{\partial g_n}{\partial a_n} \end{pmatrix}. \end{align*}$$ Do we simply multiply through $g$ and $f$ and add the two?



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