Shortcut Method For Derivative Of Polynomials

Polynomial functions take the general form:

P(x)=a_nxⁿ+a_n-1x^n-1+a_n-2x^n-2+...+a₀,

where each a term is a integer and n is the degree of the polynomial.

The definition of a derivative is

Lim        f(x+h)-f(x)
h->0   -----------------
                  h


This can be very exhausting, inefficient and even in some cases not a feasible method find derivatives of polynomials. Luckily, we have the shortcut method as follows:

for any term axⁿ, the derivative will be (n)(a)x^n-1.  For instance, if we have the term

6x¹²

the derivative is (12)(6)x^12-1 =   (12)(6)X¹¹= 72X¹¹.  In this manner, the task of finding the derivative of a polynomial is reduced to taking the derivative of each term.  For instance:


f(x)= 5x⁷+6x⁶+3x⁴+2x³+12x²+16x+5

This would have the derivative:

f'(x)=35x⁶+36x⁵+12x³+6x+24X+16.  

Here, note that if x has power 1, that is, we see just "Xx with no power on it, then, the derivative of it will be just the coefficient of x.  I.e. The derivative of the term "16x" is "16".  In addition, the derivative of a constant term, i.e. "5" in this example, is 0.  This is the case in taking derivatives for any polynomial and this is the shortcut method to doing derivatives of polynomial functions.