Derivatives of Polynomials

Suggested Prerequisites: Definition of differentiation,

Polynomials are some of the simplest functions we use. We need to know the derivatives of polynomials such as x4+3x, 8x2+3x+6, and 2. Let's start with the easiest of these, the function y=f(x)=c, where c is any constant, such as 2, 15.4, or one million and four (106+4). It turns out that the derivative of any constant function is zero. This makes sense if you think about the derivative as the slope of a tangent line. To use the definition of a derivative, with f(x)=c,

For completeness, now consider y=f(x)=x. This is the equation of a straight line with slope 1, and we expect to find this from the definition of the derivative. We are not disappointed:

Two things to note in the above: