So What Exactly Does a Derivative Tell Me?
A derivative is simply a rate of change. If f(x) is a position function, then f'(x) tells us the velocity, or the rate at which position changes. The second derivative of f(x) (f"(x)) tells us acceleration. f"(x) is also the first derivative of velocity, and describes the rate of change of velocity. Or another way to put it is the rate of change of the rate of change.
Let's examine velocity. We know that if a function is increasing on an interval, then the velocity has to be positive on that domain. Graphically, the velocity will be above the x-axis. Conversely, if a function is decreasing, then the velocity is negative and below the x-axis.
Great, Now How Does This Help Us?
Knowing the derivative of a function gives us a lot of information about that function. For example, the first derivative tells us where the extrema of a function are, or the points at which their maximum and minimum values lie. We call this the first derivative test. So how exactly does the first derivative test work? First, start off by setting f'(x) = 0, and solving for x. These values of x are called critical values.
So how do we know where the maximum and minimum values lie on f(x)? We know that if f(x) is decreasing, then f'(x) is negative; and if f(x) is increasing, then f'(x) is positive. So where f'(x) transitions from negative to positive, we have a minimum; and when f'(x) transitions from positive to negative, we have a maximum.
The second derivative, which describes the acceleration of f(x), also provides invaluable information about the function. From the second derivative, we can determine the maximum and minimum values for the velocity, which are called points of inflection. We refer to the changes that occur at these points as changes in concavity. That is, when f"(x) < 0, the function is concave down; and when f"(x) > 0, the function is concave up.
In the image below, the hump to the left of the y-axis is concave-down, and the hump to the right axis is concave up. We can see the function initially decreasing until it reaches the maximum (think first derivative test), and then continuing to decelerate at a slower rate until it reaches its minimum, at which point it increases again.
Based on this information found in the second derivative, we can use the second derivative test to determine where local extrema lie on a function. Our precondition for the second derivative test is that f'(x) = 0 for our given point x. Based on this, we have the following conditions:
-f"(x) > 0: Local Minimum
-f"(x) < 0: Local Maximum
-f"(x) = 0: Inconclusive
Let's break these conditions down a little more. First, we know that f'(x) = 0. So if our acceleration is negative at x, that means there is a maximum because the function is going to change from increasing velocity to decreasing velocity at x. Similarly when f"(x) > 0, there is a minimum because the function changed from decreasing to increasing at x. Our last condition, f"(x) = 0, is inconclusive using the second derivative test because it is a possible point of inflection. You will have to test x using the first derivative test instead.
L'Hospital's Rule
L'Hospital's rule helps us evaluate limits in an indeterminant form by taking the derivative of the expression. Some indeterminant forms include 0 * infinity, +-infinity, 0^0, 1^infinity, etc. Formally, the rule is: lim(x-->a) f(x)/g(x) = lim(x-->a) f'(x)/g'(x).
Let's take a look at a couple of examples:
1) lim(x-->0) sin(x)/x
Starting off, this limit looks like infinity, which is an indeterminant form. If you have memorized this as a "special limit", you may recognize it as 1. We can prove this using L'Hospital's rule, though. So let's take the derivatives of the top and bottom functions. When we do this, we get: lim(x-->0) cos(x)/1, which evaluates to 1.
2) lim(x-->1) 2ln(x)/(x-1)
Again, we see an indeterminant form here of 0/0. So applying L'Hospital's rule to this limit, we can evaluate lim(x-->1) (2/x)/1 = lim(x-->1)2/x. This is much easier to evaluate, and again we get 2.
Conclusion
This concludes my second tutorial on derivatives. When working with particle motion, remember to always be thinking about what velocity is needed to change a particle's path, and what acceleration is needed to change the velocity. Accounting for these variables programatically and otherwise can make your life a lot easier when describing a particle.
MultiQuote
| 