Rules for Complex Derivatives: Carathéodory's Criterion, Continuity, Restriction, Constants, Sums, Products, Quotients, the Chain Rule and Powers

In the setting of Complex Analysis: Standing Notation, let $U$ be an open subset of $(\mathbb{C},d_{\mathbb{C}})$, let $z\in U$, let $f,g:U\to\mathbb{C}$ be functions, and let $\lambda\in\mathbb{C}$. Complex differentiability and complex derivatives are as in The Complex Derivative at a Point (ii), and continuity at a point relative to a subset of $\mathbb{C}$ is as in Continuous Map Between Metric Spaces (i), for $d_{\mathbb{C}}$. Sums, constant multiples, products and quotients of functions $U\to\mathbb{C}$ are defined pointwise.

(i) Carathéodory's criterion. Let $L\in\mathbb{C}$. Then $f$ is complex differentiable at $z$ with $f'(z)=L$ if and only if there is a function $\varphi:U\to\mathbb{C}$, continuous at $z$ relative to $U$, with $\varphi(z)=L$ and

$$f(w)-f(z)=\varphi(w)\,(w-z)\qquad(w\in U).$$

(ii) Continuity. If $f$ is complex differentiable at $z$, then $f$ is continuous at $z$ relative to $U$.

(iii) Restriction. Let $W\subseteq U$ be open in $(\mathbb{C},d_{\mathbb{C}})$ with $z\in W$. If $f$ is complex differentiable at $z$, then the restriction $f|_{W}:W\to\mathbb{C}$ is complex differentiable at $z$, and $(f|_{W})'(z)=f'(z)$.

(iv) Constants. For every $c\in\mathbb{C}$, the constant function $U\to\mathbb{C}$ with value $c$ is complex differentiable at $z$ with derivative $0$.

(v) The identity. The function $U\to\mathbb{C}$, $w\mapsto w$, is complex differentiable at $z$ with derivative $1$.

(vi) Sums. If $f$ and $g$ are complex differentiable at $z$, then so is $f+g$, and $(f+g)'(z)=f'(z)+g'(z)$.

(vii) Constant multiples. If $f$ is complex differentiable at $z$, then so is $\lambda f$, and $(\lambda f)'(z)=\lambda f'(z)$.

(viii) Products. If $f$ and $g$ are complex differentiable at $z$, then so is $fg$, and

$$(fg)'(z)=f'(z)\,g(z)+f(z)\,g'(z).$$

(ix) Reciprocals. If $g(w)\neq0$ for every $w\in U$ and $g$ is complex differentiable at $z$, then so is $1/g$, and

$$\Bigl(\frac{1}{g}\Bigr)'(z)=-\frac{g'(z)}{g(z)^{2}}.$$

(x) Quotients. If $g(w)\neq0$ for every $w\in U$ and $f$ and $g$ are complex differentiable at $z$, then so is $f/g$, and

$$\Bigl(\frac{f}{g}\Bigr)'(z)=\frac{f'(z)\,g(z)-f(z)\,g'(z)}{g(z)^{2}}.$$

(xi) The chain rule. Let $V$ be an open subset of $(\mathbb{C},d_{\mathbb{C}})$ with $f(w)\in V$ for every $w\in U$, and let $h:V\to\mathbb{C}$ be a function. If $f$ is complex differentiable at $z$ and $h$ is complex differentiable at $f(z)$, then $h\circ f:U\to\mathbb{C}$ is complex differentiable at $z$, and

$$(h\circ f)'(z)=h'\bigl(f(z)\bigr)\,f'(z).$$

(xii) Powers. For every $n\in\mathbb{N}_{0}$, the function $U\to\mathbb{C}$, $w\mapsto w^{n+1}$, is complex differentiable at $z$ with derivative $(n+1)\,z^{n}$.