Product Rule for Derivatives

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Theorem

Let $\map f x, \map j x, \map k x$ be real functions defined on the open interval $I$.

Let $\xi \in I$ be a point in $I$ at which both $j$ and $k$ are differentiable.


Let $\map f x = \map j x \, \map k x$.


Then:

$\map {f'} \xi = \map j \xi \, \map {k'} \xi + \map {j'} \xi \, \map k \xi$


It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then:

$\forall x \in I: \map {f'} x = \map j x \, \map {k'} x + \map {j'} x \, \map k x$


Using Leibniz's notation for derivatives, this can be written as:

$\map {\dfrac \d {\d x} } {y \, z} = y \dfrac {\d z} {\d x} + \dfrac {\d y} {\d x} z$

where $y$ and $z$ represent functions of $x$.


General Result

Let $f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)$ be real functions differentiable on the open interval $I$.

then:

$\forall x \in I: \displaystyle D_x \left({\prod_{i \mathop = 1}^n f_i \left({x}\right)}\right) = \sum_{i \mathop = 1}^n \left({D_x \left({f_i \left({x}\right)}\right) \prod_{j \mathop \ne i} f_j \left({x}\right)}\right)$


Proof

\(\displaystyle \map {f'} \xi\) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\map j {\xi + h} \, \map k {\xi + h} - \map j \xi \, \map k \xi} h\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\map j {\xi + h} \, \map k {\xi + h} - \map j {\xi + h} \, \map k \xi + \map j {\xi + h} \, \map k \xi - \map j \xi \, \map k \xi} h\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \paren {\map j {\xi + h} \frac {\map k {\xi + h} - \map k \xi} h + \frac {\map j {\xi + h} - \map j \xi} h \map k \xi}\)
\(\displaystyle \) \(=\) \(\displaystyle \map j \xi \, \map {k'} \xi + \map {j'} \xi \, \map k \xi\)


Note that $\map j {\xi + h} \to \map j \xi$ as $h \to 0$ because, from Differentiable Function is Continuous‎, $j$ is continuous at $\xi$.

$\blacksquare$


Also see


Historical Note

The Product Rule for Derivatives was first obtained by Gottfried Wilhelm von Leibniz in $1677$.


Sources