(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|
The previous example could be written like this:
y = x3
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: dsdt
And we know you are doing 10 m per second, so dsdt = 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 dt is changing over time!
|We could write it like this:||
But it is usually written d2s dt2
Your speed increases by 4 m/s over 2 seconds, so d2s dt2 = 42 = 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.)
|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.