Introduction to Derivatives

It is all about slope!

Slope = Change in Y Change in X

gradient

We can find an average slope between two points.

 average slope = 24/15

But how do we find the slope at a point?

There is nothing to measure!

 slope 0/0 = ????

But with derivatives we use a small difference ...

... then have it shrink towards zero.

 slope delta y / delta x

Let us Find a Derivative!

To find the derivative of a function y = f(x) we use the slope formula:

Slope = Change in Y Change in X = Δy Δx

slope delta x and delta y

And (from the diagram) we see that:

x changes from x to x+Δx
y changes from f(x) to f(x+Δx)

Now follow these steps:

  • Fill in this slope formula: Δy Δx = f(x+Δx) − f(x) Δx
  • Simplify it as best we can
  • Then make Δx shrink towards zero.

Like this:

Example: the function f(x) = x2

We know f(x) = x2 , and we can calculate f(x+Δx) :

Start with: f(x+Δx) = (x+Δx)2
Expand (x + Δx)2: f(x+Δx) = x2 + 2x Δx + (Δx)2

The slope formula is: f(x+Δx) − f(x) Δx

Put in f(x+Δx) and f(x): x2 + 2x Δx + (Δx)2 − x2 Δx

Simplify (x2 and −x2 cancel): 2x Δx + (Δx)2 Δx

Simplify more (divide through by Δx): = 2x + Δx

Then, as Δx heads towards 0 we get: = 2x

Result: the derivative of x2 is 2x

In other words, the slope at x is 2x

We write dx instead of "Δx heads towards 0".

And "the derivative of" is commonly written d dx like this:

d dx x2 = 2x
"The derivative of x2 equals 2x"
or simply "d dx of x2 equals 2x"


So what does d dx x2 = 2x mean?

slope x^2 at 2 is 4

It means that, for the function x2, the slope or "rate of change" at any point is 2x.

So when x=2 the slope is 2x = 4, as shown here:

Or when x=5 the slope is 2x = 10, and so on.

Note: f'(x) can also be used for "the derivative of":

f'(x) = 2x
"The derivative of f(x) equals 2x"
or simply "f-dash of x equals 2x"

Let's try another example.

Example: What is d dx x3 ?

We know f(x) = x3 , and can calculate f(x+Δx) :

Start with: f(x+Δx) = (x+Δx)3
Expand (x + Δx)3: f(x+Δx) = x3 + 3x2 Δx + 3x (Δx)2 + (Δx)3

The slope formula: f(x+Δx) − f(x) Δx

Put in f(x+Δx) and f(x): x3 + 3x2 Δx + 3x (Δx)2 + (Δx)3 − x3 Δx

Simplify (x3 and −x3 cancel): 3x2 Δx + 3x (Δx)2 + (Δx)3 Δx

Simplify more (divide through by Δx): 3x2 + 3x Δx + (Δx)2

Then, as Δx heads towards 0 we get: 3x2

Result: the derivative of x3 is 3x2

Have a play with it using the Derivative Plotter.

Derivatives of Other Functions

We can use the same method to work out derivatives of other functions (like sine, cosine, logarithms, etc).

Example: what is the derivative of sin(x) ?

On Derivative Rules it is listed as being cos(x)

Done.

But using the rules can be tricky!

Example: what is the derivative of cos(x)sin(x) ?

We get a wrong answer if we try to multiply the derivative of cos(x) by the derivative of sin(x) ... !

Instead we use the "Product Rule" as explained on the Derivative Rules page.

And it actually works out to be cos2(x) − sin2(x)

So that is your next step: learn how to use the rules.

Notation

"Shrink towards zero" is actually written as a limit like this:

f'(x) = lim Δx→0 f(x+Δx) − f(x) Δx

"The derivative of f equals
the limit as Δx goes to zero of f(x+Δx) - f(x) over Δx"

Or sometimes the derivative is written like this (explained on Derivatives as dy/dx):

dy dx = f(x+dx) − f(x) dx

The process of finding a derivative is called "differentiation".

You do differentiation ... to get a derivative.

Where to Next?

Go and learn how to find derivatives using Derivative Rules, and get plenty of practice: