Second Derivative

(Read about derivatives first if you don't already know what they are!)

A derivative basically gives you the slope of a function at any point.

The "Second Derivative" is the derivative of the derivative of a function. So:

  • Find the derivative of a function
  • Then take the derivative of that

A derivative is often shown with a little tick mark: f'(x)
The second derivative is shown with two tick marks like this: f''(x)

Example: f(x) = x3

  • Its derivative is f'(x) = 3x2
  • The derivative of 3x2 is 6x, so the second derivative of f(x) is:

f''(x) = 6x

A derivative can also be shown as:   dy  , and the second derivative shown as:   d2y
dx dx2

Example: (continued)

The previous example could be written like this:

y = x3

dy  = 3x2 
d2y  = 6x

Distance, Speed and Acceleration

A common real world example of this is distance, speed and acceleration:

Example: A bike race!

You are cruising along in a bike race, going a steady 10 m every second.

Distance: is how far you have moved along your path. It is common to use s for distance (from the Latin "spatium").

So let us use:

  • distance (in meters): s
  • time (in seconds): t


Speed: is how much your distance s changes over time t ...

... and is actually the first derivative of distance with respect to time:  

And we know you are doing 10 m per second, so   ds   = 10 m/s


Acceleration: Now you start cycling faster! You increase your speed to 14 m every second over the next 2 seconds.

When you are accelerating your speed is changing over time.

So   ds   is changing over time!
We could write it like this:  
d ds
But it is usually written   d2s
Your speed increases by 4 m/s over 2 seconds, so   d2s   = 4/2  = 2 m/s2


Your speed changes by 2 meters per second per second.
And yes, "per second" is used twice!
It can be thought of as (m/s)/s but is usually written m/s2


(Note: in the real world your speed and acceleration changes moment to moment, but here we assume you can hold a constant speed or constant acceleration.)


So:       Example
Distance:   s   100 m
First Derivative is Speed:  
  10 m/s
Second Derivative is Acceleration:  
  2 m/s2

And the third derivative (how acceleration changes over time) is called "Jolt" ... !

Play With It

Here you can see the derivative f'(x) and the second derivative f''(x) of some common functions:

Notice how the slope of each of those functions is the derivative plotted below it.