Derivatives of polynomials - Math Insight

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...