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) = x^{3}
 Its derivative is f'(x) = 3x^{2}
 The derivative of 3x^{2} is 6x, so the second derivative of f(x) is:
f''(x) = 6x
A derivative can also be shown as \frac{dy}{dx} , and the second derivative shown as \frac{d^{2}y}{dx^{2}}
Example: (continued)
The previous example could be written like this:
y = x^{3}
\frac{dy}{dx} = 3x^{2}
\frac{d^{2}y}{dx^{2}} = 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: \frac{ds}{dt}
And we know you are doing 10 m per second, so \frac{ds}{dt} = 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 \frac{ds}{dt} is changing over time!
We could write it like this: 


dt 
But it is usually written \frac{d^{2}s}{dt^{2}}
Your speed increases by 4 m/s over 2 seconds, so \frac{d^{2}s}{dt^{2}} = \frac{4}{2} = 2 m/s^{2}
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/s^{2}
(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:  \frac{ds}{dt}  10 m/s 
Second Derivative is Acceleration:  \frac{d^{2}s}{dt^{2}}  2 m/s^{2} 
The third derivative (how acceleration changes over time) is called "Jerk" or "Jolt" !
We can actually feel Jerk when we start to accelerate, apply brakes or go around corners as our body adjusts to the new forces.
Engineers try to reduce Jerk when designing elevators, train tracks, etc.
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 function is the yvalue of the derivative plotted below it.
For example, move to where the sin(x) function slope flattens out (slope=0), then see that the derivative graph is at zero. A similar thing happens between f'(x) and f''(x). Try this at different points and other functions.