Applications of Derivatives
Applications of Derivatives tests tangents and normals, increasing and decreasing intervals, maxima and minima, and simple optimisation.
Core rules
- Slope of tangent at x=a is f'(a); normal slope is -1/f'(a) when f'(a)≠0.
- f is increasing where f'(x)>0 and decreasing where f'(x)<0, on intervals where f' has constant sign.
- Critical points satisfy f'(x)=0 or f' does not exist. First derivative sign change or second derivative test classifies them.
- For closed intervals, compare values at critical points and endpoints.
Worked example
For f(x)=x^3-3x, f'(x)=3x^2-3=3(x-1)(x+1). Critical points are x=-1 and x=1. Sign chart: f' positive outside [-1,1], negative inside. So f increases on (-∞,-1) and (1,∞), decreases on (-1,1). f(-1)=2 is a local maximum and f(1)=-2 is a local minimum.
Traps
- f'(a)=0 does not guarantee an extremum; x^3 at 0 is a counterexample.
- Do not ignore endpoints in absolute maximum or minimum questions.
- The normal is perpendicular to the tangent, so its slope is the negative reciprocal.
- A sign change of f' is the reliable first-derivative test; a single value is not enough.
- Keep domain restrictions from the original problem, such as positive lengths.