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 
dx
d2y  = 6x
dx2

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.

speed 10m in 1s

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:  

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

 

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

acceleration from 10m per 1s to 14m per 1s

When you are accelerating your speed is changing over time.

So   ds   is changing over time!
dt
We could write it like this:  
d ds
dt
dt
But it is usually written   d2s
dt2
Your speed increases by 4 m/s over 2 seconds, so   d2s   = 4/2  = 2 m/s2
dt2

 

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
Measurement
Distance:   s   100 m
First Derivative is Speed:  
ds
dt
  10 m/s
Second Derivative is Acceleration:  
d2s
dt2
  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.