Derivative Definition at a Point

A derivative is the limiting gradient of a secant line. Through \((a,f(a))\) and \((a+h,f(a+h))\), the secant gradient is \(\frac{f(a+h)-f(a)}h\), where \(h\ne0\).

As \(h\to0\), the second point approaches the first and the secant approaches the tangent. Thus \[f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}h.\]

Writing \(x=a+h\) gives the equivalent point-gradient form \[f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}.\]

For the polynomial \(f(x)=x^3\), apply the definition at a general value of \(x\): \[f'(x)=\lim_{h\to0}\frac{(x+h)^3-x^3}{h}.\]

Expanding and cancelling \(h\) for \(h\ne0\) gives \[f'(x)=\lim_{h\to0}(3x^2+3xh+h^2)=3x^2.\] This is the derivative of \(x^3\) from first principles.

A derivative exists when the difference quotient has the same finite limit from both sides. In current IB AA HL examination questions, apply this definition to polynomials.