Sum Rule for Derivatives

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:

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

Let $\map {f_1} x, \map {f_2} x, \ldots, \map {f_n} x$ be real functions all differentiable.

Then for all $n \in \N_{>0}$:

$\ds \map {D_x} {\sum_{i \mathop = 1}^n \map {f_i} x} = \sum_{i \mathop = 1}^n \map {D_x} {\map {f_i} x}$

$\ds \map {f'} \xi$

$=$

$\ds \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$

Definition of Derivative

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \frac {\paren {\map j {\xi + h} + \map k {\xi + h} } - \paren {\map j \xi + \map k \xi } } h$

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \frac {\map j {\xi + h} + \map k {\xi + h} - \map j \xi - \map k \xi} h$

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \frac {\paren {\map j {\xi + h} - \map j \xi} + \paren {\map k {\xi + h} - \map k \xi} } h$

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \paren {\frac {\map j {\xi + h} - \map j \xi} h + \frac {\map k {\xi + h} - \map k \xi} h}$

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \frac {\map j {\xi + h} - \map j \xi} h + \lim_{h \mathop \to 0} \frac {\map k {\xi + h} - \map k \xi} h$

$\ds $

$=$

$\ds \map {j'} \xi + \map {k'} \xi$

Definition of Derivative

$\blacksquare$

It can be observed that this is an example of a Linear Combination of Derivatives with $\lambda = \mu = 1$.

$\blacksquare$

The was first obtained by Gottfried Wilhelm von Leibniz in $1677$.

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Derivatives: $3.3.2$
1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $5$. Differential Calculus: Appendix: Differentiation Rules: $2.$ Sum (algebraic) rule
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives