Deriving functions

Task goals

• Set up the first-principles definition f'(x) = lim(h->0) [f(x+h) - f(x)]/h correctly for a given polynomial, forming f(x+h) by substitution.
• Differentiate a linear function from first principles and recognise that the derived function is a constant (e.g. f(x) = 4x, f(x) = 7x, f(x) = -8x).
• Differentiate a simple quadratic from first principles by expanding (x+h)^2, subtracting f(x), dividing by h, and letting h tend to zero (e.g. f(x) = x^2, f(x) = 2x^2, f(x) = 3x^2).
• Show from first principles that adding or subtracting a constant term leaves the derived function unchanged (e.g. f(x) = x^2 + 5, f(x) = x^2 - 7, f(x) = 2x^2 + 3).
• Differentiate a two-term quadratic from first principles, combining the x^2 and x parts in one limit (e.g. f(x) = x^2 + 4x, f(x) = 2x^2 - 3x, f(x) = 3x^2 + 2x).
• Differentiate a full three-term quadratic from first principles, including negative leading coefficients and terms written out of standard order (e.g. f(x) = -3x^2 + 4x - 2, f(x) = 4 - 3x^2, f(x) = 7x - 2x^2, f(x) = 5 - 2x + 3x^2).
• Expand (x+h)^3 and use it to differentiate a cubic from first principles, cancelling every term that still contains h (e.g. f(x) = x^3, f(x) = 2x^3 - 3x, f(x) = x^3 - x^2 + x, f(x) = 3x^3 - 4x^2 + 2x).
• Generalise the first-principles process to functions with literal coefficients to derive standard results such as d/dx(ax + b) = a and d/dx(ax^2 + bx + c) = 2ax + b (e.g. f(x) = ax + b, f(x) = ax^2 + bx + c, f(x) = ax^3, f(x) = 2ax^2).