## 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?

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.

__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.

## 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

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!

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!