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

$\map {\paren {f + g}'} z = \map {f'} z + \map {g'} z$


Multiple Rule

$\map {\paren {w f}'} z = w \map {f'} z$


Combined Sum Rule

Let $\map {\dfrac \d {\d z} } {c f + w g}$ denote the derivative of $c f + w g$.

Then:

$\map {\map {\dfrac \d {\d z} } {c f + w g} } z = c \dfrac \d {\d z} \map f z + w \dfrac \d {\d z} \map g z$


Product Rule

$\map {\paren {f g}'} z = \map {f'} z \map g z + \map f z \map {g'} z$


Quotient Rule

For all $z \in D$ with $\map g z \ne 0$:

$\map {\paren {\dfrac f g}'} z = \dfrac {\map {f'} z \map g z - \map f z \map {g'} z} {\paren {\map g z}^2}$


Also see

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


Sources