There are just four simple facts which suffice to take the derivativeof any polynomial, and actually of somewhat more generalthings.First, there is the rule for taking the derivative of a power function which takes the $n$th power of its input. That is,these functions are functions of the form $f(x)=x^n$. The formula is$${d\over dx}x^n=n\,x^{n-1}$$That is, the exponent comes down to become a coefficient in front ofthe thing, and the exponent is decreased by $1$.The second rule, which is really a special case of this power-functionrule, is that derivatives of constants are zero:$${d\over dx}\,c=0$$for any constant $c$.The third thing, which reflects the innocuous role of constants incalculus, is that for any functions $f$ of $x$$$\;\;{d\over dx}\,c\cdot f=c\cdot\,{d\over dx}\,f$$The fourth is that for any two functions $f,g$ of $x$, thederivative of the sum is the sum of the derivatives:$${d\over dx}(f+g)={d\over dx}f+{d\over dx}g$$Putting these four things together, we can write genera...
