Polynomial First-Principles Method

For IB AA HL exam questions, the derivative definition is applied to polynomials. Write the difference quotient before using any differentiation rule.

For \(g(x)=x^2+2x\), \[\frac{g(x+h)-g(x)}{h}=\frac{2xh+h^2+2h}{h}=2x+h+2\quad(h\ne0).\]

Now take \(h\to0\) to obtain \(g'(x)=2x+2\). Substituting \(h=0\) before cancelling would divide by zero.

The later reciprocal proof is a historical-style extension. Keep the polynomial examples as the core first-principles exam practice.