Derivative (mathematics) facts for kids

A function (black) and a tangent (red). The derivative at the point is the slope of the tangent.

In mathematics, especially in a part called differential calculus, the derivative helps us understand how things change. Think of it like finding the exact speed of a car at one specific moment, not its average speed over a trip.

The derivative tells us how much a function is changing at a single point. If you look at a graph, the derivative at a point is the slope of the tangent line that just touches the graph at that point.

You'll often see the derivative written as $\tfrac{dy}{dx}$ . This means "the change in y divided by the change in x." Another common way to write it is $f'(x)$ , which means "the derivative of the function $f$ at point $x$ ."

What is a Derivative?
Derivatives of Different Functions
Properties of Derivatives
Uses of Derivatives
Related pages
Images for kids

What is a Derivative?

An animation showing how the derivative is like the "swing" of a function as its input changes.

Imagine you have a curve on a graph. If you pick two points on that curve, you can draw a straight line between them. The slope of this line tells you how much the curve changes between those two points.

Now, imagine moving those two points closer and closer together. As they get super close, the line connecting them becomes almost exactly the same as the tangent line that just touches the curve at one point.

The derivative is basically the slope of this tangent line. It's found by making the distance between two points on the graph incredibly small. In math terms, it looks like this:

f'(a)=\lim_{h\to 0}{\frac{f(a+h)-f(a)}{h}}

This formula means that as the distance h between the two x-values gets closer to zero, the slope of the line between them becomes the slope of the tangent line.

Derivatives of Different Functions

Functions can be different types, and each type has a general rule for finding its derivative.

Linear Functions

Linear functions are simple straight lines, like $y = mx + c$ . Here, m is the slope, and c is where the line crosses the y-axis.

For a straight line, the slope is always the same everywhere. So, the derivative of a linear function is just its slope.

If you have $y = x$ , the slope is 1. So, $\tfrac{d}{dx}(x) = 1$ .
If you have Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): y = x + 5 , the line is just shifted up. The slope is still 1. So, its derivative is also 1.
If you have Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): y = 3x + 2 , the slope is 3. So, its derivative is 3.

Power Functions

Power functions look like $x^a$ , where a is a number (the exponent). Their slope changes depending on where you are on the graph.

There's a simple rule for power functions:

\tfrac{d}{dx}x^a = ax^{a-1}

Let's look at some examples:

If you have Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^6 , its derivative is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 6x^{6-1} , which is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 6x^5 .
What about $f(x) = \tfrac{1}{x}$ ? This can be written as $x^{-1}$ .

* Using the rule: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): f'(x) = -1 \cdot x^{-1-1} = -1x^{-2} . * This can be written as Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -\frac{1}{x^2} .

Roots can also be written with fractional exponents. For example, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): f(x) = \sqrt[3]{x^2} is the same as Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x^\frac{2}{3} .

* Using the rule: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): f'(x) = \frac{2}{3} \cdot x^{\frac{2}{3}-1} = \frac{2}{3}x^{-\frac{1}{3}} .

Exponential Functions

Exponential functions are different because the variable x is in the exponent, like Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2^x or Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 3^{x^2} .

The general rule for an exponential function $ab^{f\left(x\right)}$ is:

\frac{d}{dx}\left( ab^{ f\left( x \right) } \right) = ab^{f(x)} \cdot f'\left(x \right) \cdot \ln(b)

Here, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \ln(b) is the natural logarithm of b.

Let's try an example: Find the derivative of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 3 \cdot 2^{3x^2} .

Here, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a = 3 , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): b = 2 , and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): f(x) = 3x^2 .
First, find the derivative of $f(x)$ , which is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): f'(x) = 6x .
Now, put it all into the rule:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{d}{dx} \left(3 \cdot 2^{3x^2} \right) = 3 \cdot 2^{3x^2} \cdot 6x \cdot \ln \left( 2 \right) This can be rearranged to: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \ln \left(2 \right) \cdot 18x \cdot 2^{3x^2}

Logarithmic Functions

The derivative of a logarithm is often a reciprocal. The most common logarithm in calculus is the natural logarithm, written as $\ln(x)$ .

\frac{d}{dx}\ln(x) = \frac{1}{x}

Example: Find the derivative of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \ln\left(\frac{5}{x}\right) .

Using logarithm rules, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \ln\left(\frac{5}{x}\right) can be split into Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \ln(5) - \ln(x) .
The derivative of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \ln(5) is 0, because Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \ln(5) is just a constant number.
The derivative of $\ln(x)$ is $\tfrac{1}{x}$ .
So, the derivative of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \ln\left(\frac{5}{x}\right) is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 0 - \frac{1}{x} = -\frac{1}{x} .

For logarithms with a different base, like Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \log_{10}(x) , you can convert them to natural logarithms:

\frac{d}{dx} \log_{10}(x) = \frac{d}{dx} \frac{\ln{x}}{\ln{10}} = \frac{1}{\ln{10}} \frac{d}{dx} \ln{x} = \frac{1}{x \ln(10)}

Trigonometric Functions

Trigonometric functions like sine and cosine also have derivatives. These rules apply when angles are measured in radians.

The derivative of $\sin(x)$ is $\cos(x)$ .

: $\frac{d}{dx}\sin(x) = \cos(x)$

The derivative of $\cos(x)$ is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -\sin(x) .

: $\frac{d}{dx}\cos(x) = -\sin(x)$

The derivative of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sec(x) is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sec(x)\tan(x) .

: $\frac{d}{dx}\sec(x) = \sec(x)\tan(x)$

Properties of Derivatives

When you have a function made of several parts, you can often find the derivative of each part separately and then add or subtract them. This makes finding derivatives much easier!

For example, let's find the derivative of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 3x^6 + x^2 - 6 . You can break it down:

\frac{d}{dx}(3x^6) + \frac{d}{dx}(x^2) - \frac{d}{dx}(6)

The derivative of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 3x^6 is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 6 \cdot 3x^5 = 18x^5 .
The derivative of $x^2$ is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2x^1 = 2x .
The derivative of a constant number like 6 is always 0, because constants don't change.

So, the full derivative is:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): = 18x^5 + 2x - 0

= 18x^5 + 2x\,

Uses of Derivatives

Derivatives are super useful in many areas of math and science!

Finding Maxima and Minima: Imagine a roller coaster track. At the very top of a hill or the very bottom of a valley, the track is momentarily flat. This means its slope (and thus its derivative) is zero. We can use derivatives to find these highest and lowest points of a function.
Newton's Method: Derivatives help us find where a function crosses the x-axis (its "roots" or "zeros"). This is used in computer programs to solve complex equations.
Increasing or Decreasing: A derivative can tell you if a function is going up (increasing) or going down (decreasing) at any given point. If the derivative is positive, the function is increasing. If it's negative, the function is decreasing.

Images for kids

This animation shows the derivative at different points of a function.

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