Complex derivative

The complex derivative of a complex function $f$ at $z_0$ is

$f^{\prime }(z_0)={\lim }_{\Delta z\to 0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z},$

provided the limit exists. The definition looks identical to the real-variable case. The decisive difference: $\Delta z$ is a complex number, and the limit must give the same value along every direction of approach in the plane.

This requirement is much stronger than two-dimensional real differentiability of the components. It forces a deep regularity on the functions that pass: they turn out to be infinitely differentiable, equal to their Taylor series, and rigid in ways real functions never are.

A computation that succeeds

$f(z)=z^2$. Difference quotient:

$\frac{(z+\Delta z{)}^2-z^2}{\Delta z}=\frac{2z\Delta z+(\Delta z{)}^2}{\Delta z}=2z+\Delta z\to 2z.$

The limit is $2z$ regardless of how $\Delta z$ approaches $0$. So $f^{\prime }(z)=2z$ everywhere — same as the real-variable formula.

A function with no complex derivative anywhere

$f(z)=\bar{z}$. Difference quotient:

$\frac{\overline{z+\Delta z}-\bar{z}}{\Delta z}=\frac{\overline{\Delta z}}{\Delta z}.$

Along the real axis ($\Delta z=\Delta x$, real): the quotient equals $1$.

Along the imaginary axis ($\Delta z=i\Delta y$): the quotient equals $\overline{i\Delta y}/(i\Delta y)=-i\Delta y/(i\Delta y)=-1$.

Different directional limits, so $\bar{z}$ is complex differentiable nowhere.

This is striking because $\bar{z}$ is smooth in every real-variable sense — $u(x,y)=x$ and $v(x,y)=-y$ are linear and infinitely differentiable. The function fails complex differentiability not because it’s badly behaved as a function of $(x,y)$, but because its behavior is orientation-reversing (a reflection), while complex differentiability requires the function to behave locally like rotation-and-scaling.

More diagnostic failures

$f(z)=\mathrm{Re}(z)=x$. Difference quotient $\mathrm{Re}(\Delta z)/\Delta z$. Along reals: $1$. Along imaginaries: $0$. Differentiable nowhere.

$f(z)=|z{|}^2+iz$. Write $f=u+iv$ with $u=x^2+y^2-y$ and $v=x$. Computing along the real and imaginary axes:

• Along $\Delta z=h$ real: the difference quotient is $z+\bar{z}+h+i\to 2x+i$.
• Along $\Delta z=ih$ imaginary: the quotient is $-z+\bar{z}+i+\cdots \to -2iy+i=i(1-2y)$.

These agree only when $2x+i=i(1-2y)$, i.e., when $x=0$ and $y=0$. So $f$ is differentiable only at the origin, and not analytic anywhere — analyticity requires differentiability on a whole open set, but a single point isn’t open.

Differentiation rules

Despite the stringent condition, the standard rules carry over for differentiable $f,g$ and complex constant $c$:

Rule	Statement
Sum	$(f+g{)}^{\prime }=f^{\prime }+g^{\prime }$
Constant multiple	$(cf{)}^{\prime }=c{f}^{\prime }$
Product	$(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$
Quotient	$(f/g{)}^{\prime }=(f^{\prime }g-f{g}^{\prime })/g^2$
Chain	$(f(g(z)){)}^{\prime }=f^{\prime }(g(z))g^{\prime }(z)$
Power	$(z^n{)}^{\prime }=n{z}^{n-1}$

Proofs don’t use that the variable is real — only that $\mathbb{C}$ has field structure. So polynomials, rational functions, $e^z$, $\sin z$, $\cos z$ all differentiate as in real calculus, with the same formulas.

The genuinely complex content is hidden in the question: is the function differentiable to begin with? Polynomials and rationals are; products, quotients, compositions of differentiables are; but plenty of perfectly smooth-looking real-variable functions ($\bar{z}$, $|z|$, $\mathrm{Re}(z)$, $\mathrm{Im}(z)$) are not, anywhere.

The test: Cauchy–Riemann

The clean test for complex differentiability is the Cauchy-Riemann equations. Writing $f=u+iv$, $f$ is complex differentiable at $z_0$ iff $u_x=v_y$ and $u_y=-v_x$ at $z_0$, plus continuity of the partials in a neighborhood. The derivative is then $f^{\prime }=u_x+i{v}_x=v_y-i{u}_y$.

Differentiable on an open set: analyticity

A function differentiable on an open neighborhood of $z_0$ is analytic at $z_0$. The big payoffs of complex analysis — Cauchy’s theorem, the Cauchy integral formula, the residue theorem — are statements about analytic functions, not merely differentiable ones. See Analytic function.