The study of instantaneous rates of change of functions, Differential Calculus, is one of the two main branches of Calculus. Discovered in the 17th century, derivatives are based on finding the slope of any function (curve) at any point along the graph of the curve. Derivatives provided a tool that allowed scientists to understand the movements of the planets and the forces of gravity.
Before we start our study of derivatives, let's look at the average speed or velocity as you travel from Capitan to Albuquerque. The trip is about 150 miles each way. If you can imagine that it took us 3 hours to make the trip, then we can calculate the average speed or velocity:
average speed = total distance divided by total time = 150 miles/3hours = 50 miles per hour
For anyone who has ever driven in New Mexico, you know that there are large expanses of open space - no people, mostly straight roads, few law enforcement personnel. Given this reality, there are not many times on the trip when you would actually be driving at exactly 50 mph. As you start from Capitan, you are probably driving for a few miles on dirt roads. Unless you want to permanently glue road dust to your car, you will drive pretty slow. Once on the hiway, you have a higher speed limit, but you are climbing steeply up to Apache Summit, and my poor old car doesn't go too fast. Going downhill, you can really speed up, but have to be careful because of the sharp curves in the mountains. As you drive into Carizozo, you better slow down and strictly obey the speed limit - they have good cops with great radar guns. After Carizozo, you have a straight shot heading west, but the road is pretty curvy and there is a lot of climbing.
I always make a quick stop in San Antonio for ice cream and the rest area north of Socorro is a good place to stretch your legs. I can make up time once I get on the Interstate, so after all that fooling around, it usually takes me 3 hours for the trip. My average speed is 50 mph, but there is rarely a time when I am actually driving at 50 mph. If you made a graph of your trip, with time on the bottom and distance from Capitan on the y-axis, you would have a graph of the function. The slope from time t=0 to t=3 hours equals your average speed (50 mph). That means the graph has an average slope of 50 - pretty neat isn't it?
As you make the trip, you can look down at your speedometer and see your instantaneous rate of speed. Obviously, this speed changes a lot during your trip. During the times when you are stopped, time keeps moving, but your speed is zero. On the graph, this is a horizontal segment with a slope of zero. On the freeway, your speed legally get to 75 mph. Using your graph, you can find the instantaneous speed at any time by finding the slope of the curve at that exact time. This instantaneous slope is called the derivative of the function at time t.
Definition: The derivative of a curve y = f(x) is a function f'(x) (read as f prime or derivative of f) such that:
|provided the limit exists|
The value of h is a very small number that is bigger than zero. The denominator of the definition is the change in x, and the numerator is the change in y. Thus, the derivative is simply the slope of the line tangent to the curve of the function at point x.
The limit (abbreviated "lim") is a mathematical method where we let h (for example in the definition above) or any other variable get closer and closer to the number you are aiming for but it never quite gets exactly to that number. In the case of derivatives, we sneak up on zero, but we never actually get there. This is because we are not allowed to divide by zero, but we can divide by "almost" zero. In my Calculus class we spend several weeks learning about slopes, tangent lines, and limits to get ready for derivatives.
There are many ways to indicate a derivative. In the definition above, we used f'(x) to indicate the derivative of the function f(x). Here are some other ways our textbook and others refer to derivatives:
The notation you use when solving a problem depends on the type of problem you are solving. Some abbreviations are easier to use than others. The notation originally used by Newton was not easy to use in problem solving. Luckily, Liebnitz and Maria of Agnesi took Newton's original notation and modified it to what we use today.
In some problems we are interested in the derivative of a derivative. That is because each derivative is simply a function, and we can use the definition to find the derivative of most functions. These are called higher order derivatives. For our speed problem above, the first function give distance from Capitan, the derivative gives the instantaneous speed at any time on our trip, and the derivative of the derivative (called the second derivative) is the acceleration at any time on the trip. Here are examples of the first four derivatives of a simple cubic function:
|Function||y =||2x3 - 4x2 + x - 6||a cubic function|
|First derivative||y' =||6x2 - 8x + 1||a quadratic function|
|Second derivative||y'' =||12x - 8||a linear function|
|Third derivative||y''' =||12||a constant function|
|Fourth derivative||y(4) =||0||this and all higher derivatives in this example are equal to zero|
For straight lines (linear functions) the slope is constant at all points along the line. Notice in the example above - the second derivative is a straight line with a slope of 12. The third derivative is the slope of the function above it.
We learn in Algebra I how to find the slope for straight lines, but for any other shape the slope is constantly changing because the function curve is constantly changing. Calculus uses derivatives to find the equation that defines the changing slope for continuous functions. Because the definition above is quite hard to use, we have developed a set of rules to help find the derivative. In all the rules, u and v are any differentiable functions.
|d(5) = 0|
|d(x4) = 4x3|
|d(4x2) = 4d(x2) = 4(2x) = 8x|
|Sum and Difference||
|d(x3- x2) = d(x3)- d(x2) = 3x2- 2x|
|Quotient Rule for|
A good table of basic trig functions can be found at the Harvey Mudd College Caluclus tutorial site. This link also includes basic logarithmic and exponential functions, as well as using the chain rule to solve more complicated composite functions (functions of functions).
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