finding the derivative of a polynomial

Question

I wondering symbolically how you would parse a polynomial into a function and return the derivative. What data structure would I use or method to parse the polynomial? Preferably without using any libraries, as this question could pop up in a technical interview.

polynomial-> of nth degree

def derivative(polynomial):
    return derivative

Example:

f(x)  = 2x^2+3x+1
f'(x) = 4x+3

I don't want a solution, this is not homework but a hint of where I would start.

Answer 1

If the polynomial equation is given as a list. For eg: p=[9,4,12,2] return value will be [27, 8, 12]

     def derivative(p):
        derv=[]
        for i in range(len(p)-1):
            derv.append((len(p)-i-1)*p[i])
        return derv

Answer 2

A polynomial in a single variable can be represented simply as an array containing the coefficients. So for example 1 + 5x³ - 29x⁵ can be expressed as [1, 0, 0, 5, 0, -29]. Expressed in this form the derivative is easy to compute.

suppose poly is a python list as above. Then

deriv_poly = [poly[i] * i for i in range(1, len(poly))]

For sparse polynomial other representations are relatively easy, such as a list of pairs (coefficient, exponent) or dictionary mapping exponents to coefficients.

Parsing the expression is more complicated but using various parsing frameworks should be easy because the grammar is comparatively simple.