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10/17/2014 Introduction to Derivativeshttp://www.mathsisfun.com/calculus/derivatives-introduction.html 1/6
Introduction to Derivatives
It is all about slope!
Slope
= Change in YChange in X
We can find an
average
slope between two points. But how do we find the slope
at a point
?There is nothing to measure! But with derivatives we use a small difference ...
10/17/2014 Introduction to Derivativeshttp://www.mathsisfun.com/calculus/derivatives-introduction.html 2/6
... then have it
shrink towards zero
.
Let us Find a Derivative!
We will use the slope formula:
Slope
= Change in Y =
Δ
yChange in X
Δ
xto find the derivative of a function
y = f(x)
And follow these steps:ã Fill in this slopeformula:
Δy
=
f(x+Δx) − f(x)ΔxΔx
ã Simplify it as best we can,ã Then make
Δx
shrink towards zero.Here we go:
Example: the function
f(x) = x
2
We know
f(x) = x
2
, and can calculate
f(x+Δx)
:Start with:
f(x+Δx) = (x+Δx)
2
Expand (x + Δx)
2
:
f(x+Δx) = x
2
+ 2x Δx + (Δx)
2
10/17/2014 Introduction to Derivativeshttp://www.mathsisfun.com/calculus/derivatives-introduction.html 3/6
Start with the slope formula:f(x+Δx) − f(x)ΔxPut in
f(x+Δx)
and
f(x)
:x
2
+ 2x Δx + (Δx)
2
− x
2
ΔxSimplify (x
2
and −x
2
cancel): = 2x Δx + (Δx)
2
ΔxSimplify more (divide through by Δx):
= 2x + Δx
And then
as Δx heads towards 0
we get:
= 2x
Result: the derivative of
x
2
is
2x
We write
dx
instead of
Δx heads towards 0
, so the derivative of is commonlywritten
x
2
= 2x
The derivative of
x
2
equals
2x
or simply d dx of
x
2
equals
2x
What does
x
2
= 2x
mean?
It means that, for the function x
2
, 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.
10/17/2014 Introduction to Derivativeshttp://www.mathsisfun.com/calculus/derivatives-introduction.html 4/6
Note: sometimes
f’(x)
is also used for the derivative of :
f’(x) = 2x
The derivative of
f(x)
equals
2x
Let's try another example.
Example: What is
x
3
?
We know
f(x) = x
3
, and can calculate
f(x+Δx)
:Start with:
f(x+Δx) = (x+Δx)
3
Expand (x + Δx)
3
:
f(x+Δx) = x
3
+ 3x
2
Δx + 3x (Δx)
2
+ (Δx)
3
The slope formula:f(x+Δx) − f(x) ΔxPut in
f(x+Δx)
and
f(x)
:x
3
+ 3x
2
Δx + 3x (Δx)
2
+ (Δx)
3
− x
3
ΔxSimplify (x
3
and −x
3
cancel): = 3x
2
Δx + 3x (Δx)
2
+ (Δx)
3
ΔxSimplify more (divide through by Δx):
= 3x
2
+ 3x Δx + (Δx)
2
And then
as Δx heads towards 0
weget:
x
3
= 3x
2
Have a play with it using the Derivative Plotter.

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