limit as h approaches 0: f(z+h)-f(z) over h
This time h is complex, a vector with two real components. Let h approach 0, using the notion of distance in the complex plane, and the difference quotient must approach f′(z). This is the standard interpretation of the limit of a vector function.
The limit of the sum is the sum of the limits, for real or vector functions, hence the derivative of f+g is f′+g′.
Multiply f(z) by a constant c and the difference quotient is multiplied by c. This multiplies the limit by c, even if c is complex. The derivative of a linear combination of functions is the linear combination of their derivatives.
Those who wish to be rigorous may want to review the theorems on limits, products, and quotients. The limit of the product or quotient is the product or quotient of the limits, even for complex functions. With this in mind, you can apply the proofs you have already seen for the product rule, quotient rule, chain rule, and inverse function rule. Everything is (symbolically) the same.
In particular, the derivative of a complex polynomial is computed as though it were a real polynomial. Fold the exponents into the coefficients and decrease the exponents by 1. The derivative of 3z2+7z is 6z+7.
A function is analytic at p if it is differentiable in some neighborhood containing p. A function can be analytic throughout a domain, or over the entire plane, in which case it is sometimes called "entire".
A singularity, or pole, is a point at which a function is not differentiable, even though it is differentiable in some neighborhood about that point.