Category:Combination Theorem for Complex Derivatives
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This category contains pages concerning Combination Theorem for Complex Derivatives:
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 $\map g z \ne 0$:
- $\map {\paren {\dfrac f g}'} z = \dfrac {\map {f'} z \map g z - \map f z \map {g'} z} {\paren {\map g z}^2}$
Pages in category "Combination Theorem for Complex Derivatives"
The following 13 pages are in this category, out of 13 total.
C
- Combination Theorem for Complex Derivatives
- Combination Theorem for Complex Derivatives/Combined Sum Rule
- Combination Theorem for Complex Derivatives/Multiple Rule
- Combination Theorem for Complex Derivatives/Product Rule
- Combination Theorem for Complex Derivatives/Product Rule/Proof 1
- Combination Theorem for Complex Derivatives/Product Rule/Proof 2
- Combination Theorem for Complex Derivatives/Quotient Rule
- Combination Theorem for Complex Derivatives/Sum Rule
- Combination Theorem for Complex Derivatives/Sum Rule/Proof 1
- Combination Theorem for Complex Derivatives/Sum Rule/Proof 2