> [!theorem] Product Rule > > Let $\cx_1, \cx_2, \cy$ be [[Banach Space|Banach spaces]], and $m: \cx_1 \times \cx_2 \to \cy$ be a [[Continuity|continuous]] [[Multilinear Map|bilinear map]], then > $ > Dm(x_1, x_2)(v_1, v_2) = m(x_1, v_2) + m(v_1, x_2) > $ > meaning that its [[Higher Derivatives|higher derivatives]] are constant or $0$. > > If $\cx$ is also a Banach space, $f: \cx \to \cx_2$ and $g: \cx \to \cx_2$ are [[Derivative|differentiable]] at $x$, then (omitting $m$ as the multiplication) > $ > \begin{align*} > D(fg)(x) = Df(x) \cdot g(x) + f(x) \cdot Dg(x) > \end{align*} > $ > > ### Multiplication Map > > $ > \begin{align*} > &m(x_1 + h_1, x_2 + h_2) - m(x_1, x_2) \\ > &= m(x_1, x_2 + h_2) + m(h_1, x_2 + h_2) - m(x_1, x_2) \\ > &= m(x_1, x_2) + m(x_1, h_2) \\ > &+ m(h_1, x_2) + m(h_1, h_2) - m(x_1, x_2) \\ > &= m(x_1, h_2) + m(h_1, x_2) + m(h_1, h_2) > \end{align*} > $ > where > $ > \begin{align*} > \abs{m(h_1, h_2)} &\le C \cdot \norm{h_1} \cdot \norm{h_2} \\ > \frac{\abs{m(h_1, h_2)}}{\max(\norm{h_1}, \norm{h_2})} &\le C \cdot \max(\norm{h_1}, \norm{h_2}) > \end{align*} > $ > which approaches $0$ as $\norm{(h_1, h_2)} \to 0$. Therefore $m(h_1, h_2)$ is little-oh, and > $ > Dm(x_1, x_2)(h_1, h_2) = m(x_1, h_2) + m(h_1, x_2) > $ > > ### Product Rule > > $ > \begin{align*} > &f(x + h) \cdot g(x + h) - f(x) \cdot g(x) \\ > &= f(x + h) \cdot g(x + h) - f(x + h) \cdot g(x) \\ > &+ f(x + h) \cdot g(x) - f(x) \cdot g(x) \\ > &= f(x + h) \cdot (g(x + h) - g(x)) \\ > &+ (f(x + h) - f(x)) \cdot g(x) \\ > &= f(x + h) \cdot (Dg(x)(h) + o_g(h))\\ > &+ (Df(x)(h) + o_f(h)) \cdot g(x) \\ > &= f(x + h) \cdot Dg(x)(h) + f(x + h) \cdot o_g(h) \\ > &+ Df(x)(h) \cdot g(x) + o_f(h) \cdot g(x) \\ > &= \underbrace{Df(x)(h) \cdot g(x) + f(x) \cdot Dg(x)(h)}_{\lambda} \\ > &+ \underbrace{(f(x + h) - f(x)) \cdot Dg(x)(h)}_{o_1} \\ > &+ \underbrace{f(x + h) \cdot o_g(h)}_{o_2} + \underbrace{o_f(h) \cdot g(x)}_{o_3} > \end{align*} > $ > Now, > $ > \begin{align*} > &\frac{\norm{(f(x + h) - f(x)) \cdot Dg(x)(h)}}{\norm{h}} \\ > &\le \frac{C\norm{f(x + h) - f(x)} \cdot \norm{Dg(x)} \cdot \norm{h}}{\norm{h}} \\ > &= C\norm{f(x + h) - f(x)} \cdot \norm{Dg(x)} \to 0 > \end{align*} > $ > as $h \to 0$ (since $f$ is differentiable). The two other terms have norms bounded by a constant multiple of $o_g$ or $o_f$, and are little-oh as well.