Combination Theorem for Complex Derivatives

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Theorem

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

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

Let $z \in D$.

Let $w, c \in \C$ be arbitrary complex numbers.


Then the following results hold:


Sum Rule

$\left({f + g}\right)' \left({z}\right) = f' \left({z}\right) + g' \left({z}\right)$


Multiple Rule

$\left({w f}\right)' \left({z}\right) = w f' \left({z}\right)$


Combined Sum Rule

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

Then:

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


Product Rule

$\left({f g}\right)' \left({z}\right) = f' \left({z}\right) g \left({z}\right) + f \left({z}\right) g' \left({z}\right)$


Quotient Rule

For all $z \in D$ with $g \left({z}\right) \ne 0$:

$\left({\dfrac f g}\right)' \left({z}\right) = \dfrac{ f' \left({z}\right) g \left({z}\right) - f \left({z}\right) g' \left({z}\right) }{ \left({g \left({z}\right) }\right)^2 }$


Also see

These theorems show that identical results hold for derivatives of real functions.


Sources