1.7.3—Differentiation applications

f′(x) gives the gradient or rate at x; solve f′(x)=m for a specified tangent gradient and f′(x)=0 for stationary candidates. Units carry through the rate interpretation.

Translate the derivative back into the problem: distance gives velocity, velocity gives acceleration, and a stationary candidate still needs classification or endpoint comparison for an optimum.

If s(t)=t³−6t²+9t, then v=s′=3t²−12t+9; stationary position occurs where v=0, but the physical time interval decides which roots matter.

A zero derivative is not automatically a maximum or minimum and may be a stationary inflection.