A Gentle Introduction To Function Derivative Function, Rules Of Calculus

The concept of the derivative is the building block of many topics of calculus. It is important for understanding integrals, gradients, Hessians, and much more.

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In this tutorial, you will discover the definition of a derivative, its notation and how you can compute the derivative based upon this definition. You will also discover why the derivative of a function is a function itself.

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After completing this tutorial, you will know:

The definition of the derivative of a functionHow to compute the derivative of a function based upon the definitionWhy some functions do not have a derivative at a point

Let’s get started.

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A Gentle Introduction to Function Derivatives Photo by Mehreen Saeed, some rights reserved

Tutorial Overview

This tutorial is divided into three parts; they are:

The definition and notation used for derivatives of functionsHow to compute the derivative of a function using the definitionWhy some functions do not have a derivative at a point

What is the Derivative of a Function

In very simple words, the derivative of a function f(x) represents its rate of change and is denoted by either f”(x) or df/dx. Let’s first look at its definition and a pictorial illustration of the derivative.

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Illustration of the Definition of a Function Derivative

In the figure, Δx represents a change in the value of x. We keep making the interval between x and (x+Δx) smaller and smaller until it is infinitesimal. Hence, we have the limit (Δ

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