Derivative Definition at a Point

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

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

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

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

Expanding and cancelling \(h\) for \(h\ne0\) gives \[\mathrm{f}'(x)=\lim_{h\to0}(3x^2+3xh+h^2)=3x^2.\] 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.