One of the first things you'll learn in Calculus is how to differentiate a polynomial. Basically what's going on is you're taking a polynomial and making a function to show its slope. For example:

+-----------------------------------------------------------------+ | | ## | 2.0000 | | ## | 1.8000 | | # | 1.6000 | | ## | 1.4000 | | # | 1.2000 | | # | 1.0000 | | ## | .80000 | | # | .60000 | | ## | .40000 | | ## | .20000 |----------------------------#########----------------------------| .00000 | ## | | -.2000 | ## | | -.4000 | # | | -.6000 | ## | | -.8000 | # | | -1.000 | # | | -1.200 | ## | | -1.400 | # | | -1.600 | ## | | -1.800 | # | | -2.000 +-----------------------------------------------------------------+ ^-3.095 ^-2.321 ^-1.547 ^-.7738 ^.00000 ^.77380 ^1.5476 ^2.3214 ^3.0952

This is the function *x ^{3}*

+-----------------------------------------------------------------+ | | # | 2.0000 | # | # | 1.8000 | # | ## | 1.6000 | # | # | 1.4000 | # | ## | 1.2000 | # | # | 1.0000 | # | # | .80000 | # | ## | .60000 | # | # | .40000 | ## | ## | .20000 |-------------------------------###-------------------------------| .00000 | | | -.2000 | | | -.4000 | | | -.6000 | | | -.8000 | | | -1.000 | | | -1.200 | | | -1.400 | | | -1.600 | | | -1.800 | | | -2.000 +-----------------------------------------------------------------+ ^-3.095 ^-2.321 ^-1.547 ^-.7738 ^.00000 ^.77380 ^1.5476 ^2.3214 ^3.0952

And this function (*3x ^{2}*, called the

*derivative*of

*x*) shows the slope of

^{3}*x*. In places where

^{3}*x*is increasing, the derivative (the slope) is positive. In places where

^{3}*x*is decreasing, the derivative is negative.

^{3}So we write:

*(d/dx) x ^{3} = 3x^{2}*

*d/dx* stands for "the derivative with respect to x," meaning that x is the independent variable. The above equation could be read "the derivative of *x ^{3}* is

*3x*"

^{2}So how do you figure out what the derivative for a given function is? Well, first of all, the derivative of any constant is 0. The derivatives of 1, 2, 500, and 3,000,000 are all 0. This is because if you make a graph of a constant, you get a straight horizontal line, with a slope of 0.

Next, we have the power rule, written thus: *(d/dx) x ^{n} = nx^{n-1}*. So if you have

*x*to any power, just multiply it by that power and subtract 1 from the exponent to get the derivative. For example,

*(d/dx) x*. If you have a coefficient, just leave it there. Like this:

^{5}= 5x^{4}*(d/dx) 3x*.

^{5}= 3 * 5x^{4}= 15x^{4}Just one more rule now: the sum rule. To put it in words, if you have multiple terms in your polynomial, just add up the derivatives. Like this:

*(d/dx) (5x ^{2} + 6x^{3}) = (d/dx) 5x^{2} + (d/dx) 6x^{3} = 5*2x + 6*3x^{2} = 10x + 18x^{2}*

Hopefully I was relatively clear. My intention in writing this was not to impart a thorough knowledge of Calculus, but just to describe the relatively simple procedure of differentiating a polynomial.