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 f(a+h)-f(a)h, where h 0. As h 0, the second point approaches the first and the secant approaches the tangent. Thus f'(a)=lim_h 0 f(a+h)-f(a)h. Writing x=a+h gives the equivalent point-gradient form f'(a)=lim_x to a(f(x)-f(a))/(x-a). For the polynomial f(x)=x3, apply the definition at a general value of x: f'(x)=lim_h 0((x+h)3-x3)/(h). Expanding and cancelling h for h 0 gives f'(x)=lim_h 0(3x2+3xh+h2)=3x2. This is the derivative of x3 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.