Combination Theorem for Complex Derivatives/Combined Sum Rule

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Theorem

Let $D$ be an open subset of the set of complex numbers.

Let $f, g: D \to \C$ be complex-differentiable functions on $D$.

Let $c, w \in \C$.


Let $\dfrac \d {\d z} \left({c f + w g}\right)$ denote the derivative of $c f + w g$.

Then:

$\dfrac \d {\d z} \left({c f + w g}\right) \left({z}\right) = c \dfrac \d {\d z} f \left({z}\right) + w \dfrac \d {\d z} g \left({z}\right)$

for all $z \in D$.


Proof

\(\displaystyle c \dfrac \d {\d z} f \left({z}\right) + w \dfrac \d {\d z} g \left({z}\right)\) \(=\) \(\displaystyle \dfrac \d {\d z} \left({c f}\right) \left({z}\right) + \dfrac \d {\d z} \left({w g}\right) \left({z}\right)\) Combination Theorem for Complex-Differentiable Functions: Multiple Rule
\(\displaystyle \) \(=\) \(\displaystyle \dfrac \d {\d z} \left({c f + w g}\right) \left({z}\right)\) Combination Theorem for Complex-Differentiable Functions:Sum Rule

$\blacksquare$


Sources