So you decided to derive the derivative.
Do you know?
The derivative could also mean:
1. A financial product.
2. Something that is derived.
3. A word that is made out of another one.
Before we start, we need to go over some different notations of the derivative:
1. Newton's notation: the derivative of f(x) is f '(x). It is especially good for demonstration purposes-- for example, on this site.
2. Leibniz's notation: the derivative of y=f(x) is dy/dx. Especially good for complicated manipulations-- for example, on the documents for download.
3. As an operator on a function-- the operator is marked by Dx.
Rule of the Thumb—Intuition can carry you a long way
Before we start rigorously studying derivatives, I shall first tell you how they are calculated. You have not read me wrong. Now, may hell go with the chain rule, the limit definition, and continuity. Let’s grab a polynomial function that comes to mind—how does 2x^7+5x^6+x^2+2 sound? That came off my random number generator, with coefficients generated from 0 to 10. So. What is the derivative of this mess?
RULE OF THUMB: take the exponent, move it down, multiply it with the coefficient—it’s the new coefficient. Now take one from the exponent. This is the new exponent. (This is for integer coefficient polynomials, as you can see, this is painfully limited)
So, for 2x^7, we have the derivative 14x^6. Simple enough? Another exercise: x^2. The derivative is 2x. Now ask yourself an intuitive question. What is the derivative of 2x^7+x^2? You are right. It is 14x^6+2x. We shall soon see to the rigorous part of the calculation we just made--but remember: it is the calculations that matter in the real world. The rigorous part is important, for sure, but don't let it take the fun out of math.
The laws can carry you a long way...
Properties of the derivative
1. The Power Rule
This is just the rule of the thumb above.
The proof of almost every theorem in derivatives refers back to its limit definition. Then, it's just a matter of elementary algebra. The power rule can be proved by the binomial theorem in algebra.
2. Addition and Subtraction Rules
The derivative of f(x)+g(x) is just f'(x)+g'(x). Changing g(x) to -g(x) gives us the subtraction rule.
The proof relies on addition properties of fractals. You should be familiar with that by now, or there is really nothing i could say.
3. Product Rule
(Here is where things get tricky)
The derivative of f(x)*G(x) is f'(x)*G(x)+f(x)*G'(x).
It is not: f'(x)*G'(x)! Leibniz made the same mistake, however, he had the leisure of fixing it. IF YOU MAKE THE SAME MISTAKE ON A TEST, YOU WOULDN'T ENJOY THE SAME PLEASURE.
The proof of this theorem, like all others, relies on the operations and manipulations of fractals.
4.The Quotient Rule
This follows immediately from the product rule and the power rule. It states:
The derivative of f(x)/g(x) is [f'(x)g(x)-f(x)g'(x)]/g(x)^2.
There, we have finished accounting for the four most important rules in calculating derivatives. Get them straight, and derivatives would get you straight!
Higher Degree Derivatives
All right. you have learned how to differentiate. That is, to derive the derivative. Now, suppose we do this to a function many many times.
Thus comes higher degrees derivatives. Differentiate once to get the first degree derivative, or commonly known as, the derivative. Twice, you get the second degree derivative, often noted by f''(x). Thrice, f'''(x), and four times, f(4)(x). now, (4) should be up there. You know what I mean.
It's really not difficult. wait till you get to its applications.
Climbing the staircase of the derivative.
Some Prominent Interpretations (Uses) of the Derivative
Physical Interpretation: instantaneous ... something. That's why it's so useful-- anything that is instantaneous in physics has got something to do with the derivative. Be it velocity, or acceleration, or force. (All INSTANTANEOUS!)
Economical Interpretation: Marginal... something. Example: Marginal benefit, marginal revenue...
Geometrical Interpretation: The slope of the line which is tangent to a point of a curve. It is equal to the derivative of the curve at that point.
The Important Limit
Here, we define the limit rigorously. First, we shall ask ourselves what great men asked themselves 300 years ago. As we can see, we are horribly late. We can calculate the average speed of an object. If a car travelled 132 miles in 2 hour, its average speed is 66 miles per hour. By the way, the interstate speed limit is 65, for those of you who have a car. The average speed. We hate to say that.
What if we wish to know the speed at a certain point of time? That would be the number on the speedometer at that point of time. Now, we define the “that point of time” to be a time period of length zero. What does that mean?
On one hand, it means a car cannot move an inch in a “point of time”. For distance needs time to accumulate. And those familiar with Red Alert should immediately recognize it as a trans-chronicle vehicle. On the other hand—the car is certain to have a velocity, or the speedometer is malfunctioning. How to find this instantaneous velocity? Though it sounds incredible, we divide distance by time. (Which are both zero)
Here, we witness the first marvel of calculus—making something out of nothing. Now suppose the car is constantly moving forward at 66 miles per hour. We first chose this to be the case so that we have a strong feeling about what is going to be the answer.
If its speed is always 66 miles per hour, it’s speed at “a point of time” is obviously 66 miles per hour. Now we solve it mathematically, using limits. We choose the function f(t), with time as the independent variable, measured in hours. We at once have y=66t, while y is in miles. Now suppose we plot this graph in the coordinate plane. It will be a straight line, passing through the origin. Obviously, the speed is the slope at that point.
The slope is given by [f(x+h)-f(x)]/h. Suppose we set h to 1. We calculate the slope as 65. Now what if we set h to 0? That would represent a time period with 0 length. In other words, it will represent a “point of time”. This operation is obviously illegal with regular arithmetic—the denominator cannot be 0. Yet, we are now armed with the weapon of limits—what is the limit of [f(x+h)-f(x)]/h at x=0? The answer would be the “instantaneous slope” at that point, or the tangent line to the graph at that point. That would be our instantaneous velocity.
Evaluating the limit immediately gives us that the instantaneous velocity is 66. (use the 0/0 type strategy) as we expected.
Here, we give the formal definition of the derivative:
Or, the value of the derivative of a function at a certain point is equal to the slope of the tangent line at that point.
It's really this simple. For quicker ways to calculate derivatives, please consult the power rule and the chain rule.