In calculus you learn about functions. A function is a relationship that connects each value of x to one and only one value of y. This is then denoted as y = f(x), or "y is a function of x." Many functions are continuous, i.e. there are no values of x where f(x-a) is different from f(x+a) when a is made arbitrarily small. Of those continuous functions, many are "smooth" enough to have no sharp kinks for any x. An example of a function that is not smooth at all points is the function y = |x| or y equals the absolute value of x. This function connects to each negative x the same number without the negative sign, and to each non-negative x, the same value as x. Graphically this function appears as a line with a 45-degree downwards slope for negative numbers, ending at the point x=0, y=0, also denoted as (0,0), and a line with an upwards 45-degree slope starting from (0,0). For this function there is no well-defined slope at the point (0,0). It has a slope of -1 from the left and +1 from the right.For smooth functions one can draw a single tangent at any point x. Each tangent has a well-defined slope. The relationship between x and the slope of the tangent to f(x) at x is called the derivative function and is denoted either as df(x)/dx or f'(x) (f prime of x).Polynomials are functions that have the form:f(x) = a0 + a1 * x + a2 * x^2 + a3 * x^3 + ... + an * x^n,where ak (for k = 0 ... n) are constants. Since the derivative of a sum is equal to the sum of the derivatives, you can take the derivative of each term in the polynomial by itself, and then sum up those derivatives.For this explanation of how to calculate the derivative of a polynomial we'll use as an example the polynomial:f(x) = 1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4

## Calculating the derivative of polynomials

Ignore the term a0. That term is constant and as such has a slope of 0 at all points. Therefore its derivative is 0 everywhere. In our example, a0 = 1, and its derivative is 0.

Calculate the derivative of the term, a1 * x. This is simply a1. In our example, the derivative of 2*x is simply 2.

Calculate the derivative of each of the remaining terms. For a term of the general form ak * x^k, the derivative is simply k * ak * x^(k-1). In our example, the remaining terms are 3*x^2, 4*x^3 and 5*x^4. Their derivatives are simply 2*3*x which is 6*x, 4*3*x^2 which is 12*x^2, and 4*5*x^3 which is 20*x^3.

Sum up all the derivatives of the different terms. The result is of the form:f'(x) = a1 + 2*a2 * x + 3*a3 * x^2 + 4*a4 * x^3 + ... + n*an * x^(n-1).For our example the derivative is:f'(x) = 2 + 6*x + 12*x^2 + 20*x^3