Introduction to Derivatives

It is all about slope!

Slope = Change in YChange 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 = ΔyΔx

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:

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 ddx like this:

ddxx2 = 2x
"The derivative of x2 equals 2x"
or simply "d dx of x2 equals 2x"


So what does ddxx2 = 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 ddxx3 ?

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).

But in practice the usual way to find derivatives is to use:

Derivative Rules

 

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) = limΔx→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):

dydx = 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:

 
Derivative Rules Calculus Index