Is there an easier procedure to determine derivative of a polynomial using first principles, i.e. do I have to expand the terms in brackets for f(x) = x^3 or is there an easier way.
From first principles the derivative of a function f(x) is given by: `f'(x) = lim_(h->0)(f(x+h) - f(x))/h.`
In the case of a polynomial f(x) = ax^n + bx^(n - 1)... to determine the derivative you would have to expand each of the terms that are in the form (x+h)^n, then cancel what can be and finally divide the resulting expression by h before substituting h = 0. There is no easier way to do this for a general polynomial. This is the reason why polynomials where the variable is raised to powers greater than 2 to 3 is seldom given in questions where the derivative is to be derived from first principles. After all if you have a general idea of how it is done and given a sufficient duration of time the same can be done for any polynomial.
For the polynomial f(x) = x^3, the derivative from first principles is:
`lim_(h->0) ((x+h)^3- x^3)/h`
= `lim_(h->0) (x^3 + h^3 + 3h^2x + 3x^2h- x^3)/h`
= `lim_(h->0) (h^3 + 3h^2x + 3x^2h)/h`
= `lim_(h->0) h^2 + 3hx + 3x^2`
substitute h = 0
The derivative of f(x) = x^3 is f'(x) = 3x^2