Derivative in Math – Explanation with Examples

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When discussing creative writing, saying something is “derivative” implies that there wasn’t much thought put in, and the writer copied ideas from other works.

In math, derivative equations aren’t equations that suggest a lack of imagination but rather help find the average slope between two points. Derivative math problems can range from “hey, that’s not so hard” to “when did math become all letters and symbols?”

This article will discuss how to calculate derivatives and make the concept as clear as possible. While learning math can feel like learning a new language, once you understand what new symbols mean and how they operate, you’ll be able to get some more complex equations.

If you’re wondering how finding the derivative of a function is ever going to come in handy, well, let’s say you want to get into one of the following careers:

  • Engineering
  • Construction
  • Architecture
  • Visual Arts
  • Seismology
  • Medicine
  • Computer Programming

Understanding derivatives calculus equations, particularly without the use of a calculator, will be incredibly helpful.

What Does Derivative Mean?

We’ll look at some derivative definition math examples shortly, but let’s give you a working definition for now.

The derivative shows the rate of change of functions with respect to variables.

In calculus and differential equations, derivatives are essential for finding solutions. Let’s look at a derivative math equation to better understand the concept and offer some definitions for the various symbols used.

The simplest way to look at a derivative equation is to relate it to a slope on a graph.

We can see that x runs along the bottom horizontally, and y runs along the left side vertically.

There’s a line running through the graph, and we’re going to find a number representing the overall change or average slope between two points on the line. We can write the equation this way:

Slope = Change in Y

Change in X

If the slope of the line were such that the change in x between the two points were 3, and the change in y was 6, we would get an equation that looks like this:

Slope = 6


Divide to arrive at:

Slope = 2

We now know the average slope of the line! Easy, right? Well, hold on, because what do we do if we need to find the average slope between two points on a curved line?

With curves, the formula for how to calculate derivatives gets a bit more complex. Plus, we’re going to add in our first derivative math symbol.

Slope = Change in Y = Δy

Change in X = Δx

The triangle symbol, Δ, is called “Delta.” We can think of it as meaning “change in.”

The formula would be the change in y divided by the change in x. Now we’ll get to another symbol we need to know.

Take a look at this equation:

Δy = f(x+Δx)

Δx Δx

See the f? The f in derivative math equations means the “function,” or how much the slope changes. F relates input to output so that we can understand the relationship between equation and answer. If we had a function:

f(x) = 2x

We would know that whatever number we put in for x would be multiplied by 2.

f(3) = 2(3)

Multiply, and we get:

f(3) = 6

Remember that when solving a derivative equation, the goal is to make Δx move towards zero.

You might think you’re looking at a bunch of code so far, but trust us, you’ll get used to it.

First, we’re going to take our function and apply it to our equation. Since f(x)=2x, we know that we want to use this to our advantage when solving this problem. We’ll start one piece at a time.

If f(x)=2x

Then f(x+Δx)=2(x+Δx) because essentially “f” means that we multiply whatever’s in the parentheses after f by 2.

Using the distributive property of algebra, we know that we can simplify to get 2x+2(Δx), which we’ll change to 2Δx+2x:

Δy = 2Δx+2x

Δx Δx

Because we’re working on getting Δx to be as close to zero as possible, what happens when we put 0 in for Δx? Well, we’re dividing by 0 at that point, and 2 times 0 is zero, so we’re left with this:



This step brings us to zero! So we know that whatever figure we put in for Δx, we’ll know the average rate of change for our slope.

Phina Pipia

Phina Pipia is a writer, educator, and performer. Her work can be found at

Phina Pipia

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