# Derivative of Complex Functions

Learn what is the derivative of complex functions with formula and examples. Also understand how to differentiate complex function by using partial derivative.

Alan Walker-

Published on 2023-05-26

## Introduction to the Derivative of Complex Functions

A function containing a real and an imaginary part is known as a complex function. Since calculus involves the study of rate of change, it also involves the study of rate of change of complex functions. It can be calculated by using different techniques of differentiation. Let’s understand complex function differentiation and its method of calculation. Before this, let’s discuss what complex functions are.

## What are complex functions?

The understanding of the complex functions is important to understand the complex differentiation. A complex function can be defined as a relation between two complex numbers. In other words, it is a function that takes a complex number as an input and produces a complex number as an output.

In mathematical terms, a complex number contains a real and an imaginary part which is expressed by ‘i’. It is written as;

$f(z) = u(x,y)+v(x,y)i$

Where,

## Rules of Differentiation For Complex Functions

Just like real functions, complex functions also have rules of differentiation. These rules make it easier to find the derivative of a complex function. Here are some of the rules of differentiation for complex functions:

1. Power Rule: If f(z) = z^n, where n is a constant, then f'(z) = nz^(n-1).
2. Sum Rule: If f(z) = g(z) + h(z), then f'(z) = g'(z) + h'(z).
3. Product Rule: If f(z) = g(z)h(z), then f'(z) = g(z)h'(z) + g'(z)h(z).
4. Quotient Rule: If f(z) = g(z)/h(z), then f'(z) = [g'(z)h(z) - g(z)h'(z)]/h(z)^2.
5. Chain Rule: If f(z) = g(h(z)), then f'(z) = g'(h(z))h'(z).

Since the complex differentiation also uses the same derivative rules, you can use our chain rule calculator to find the solution of a complex function derivative.

## Conclusion

In mathematics, the derivative of complex function is the rate of change of the real and imaginary parts. It is calculated by using ordinary derivative laws such as chain rule, product rule, power rule or quotient rule. For this, the Cauchy-Riemann equation is used as a formula. This derivative is used to analyse the complex problems in calculus.