Sum 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 $f$ is differentiable at $\xi$ and:
- $\map {f'} \xi = \map {j'} \xi + \map {k'} \xi$
It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then:
- $\forall x \in I: \map {f'} x = \map {j'} x + \map {k'} x$
General Result
Let $f_1 \left({x}\right), f_2 \left({x}\right), \ldots, f_n \left({x}\right)$ be real functions all differentiable.
Then for all $n \in \N_{>0}$:
- $\displaystyle D_x \left({\sum_{i \mathop = 1}^n f_i \left({x}\right)}\right) = \sum_{i \mathop = 1}^n D_x \left({f_i \left({x}\right)}\right)$
Proof 1
\(\displaystyle \map {f'} \xi\) | \(=\) | \(\displaystyle \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h\) | Definition of Derivative | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \lim_{h \mathop \to 0} \frac {\paren {\map j {\xi + h} + \map k {\xi + h} } - \paren {\map j \xi + \map k \xi } } h\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \lim_{h \mathop \to 0} \frac {\map j {\xi + h} + \map k {\xi + h} - \map j \xi - \map k \xi} h\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \lim_{h \mathop \to 0} \frac {\paren {\map j {\xi + h} - \map j \xi} + \paren {\map k {\xi + h} - \map k \xi} } h\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \lim_{h \mathop \to 0} \paren {\frac {\map j {\xi + h} - \map j \xi} h + \frac {\map k {\xi + h} - \map k \xi} h}\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \lim_{h \mathop \to 0} \frac {\map j {\xi + h} - \map j \xi} h + \lim_{h \mathop \to 0} \frac {\map k {\xi + h} - \map k \xi} h\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \map {j'} \xi + \map {k'} \xi\) | Definition of Derivative |
$\blacksquare$
Proof 2
It can be observed that this is an example of a Linear Combination of Derivatives with $\lambda = \mu = 1$.
$\blacksquare$
Historical Note
The Sum Rule for Derivatives was first obtained by Gottfried Wilhelm von Leibniz in $1677$.