Application of Derivatives
Application of Derivatives tests rates of change, tangents and normals, monotonicity, maxima and minima, and simple optimisation. The exam expects sign analysis and justification, not only formula substitution.
Core rules
The slope of the tangent to y=f(x) at (x₀,y₀) is f′(x₀); the normal slope is −1/f′(x₀), provided f′(x₀)≠0. A differentiable function is increasing where f′(x)>0 and decreasing where f′(x)<0. Critical points occur where f′(x)=0 or f′ is undefined. The first derivative test uses sign changes; the second derivative test uses f″(x₀)>0 for a local minimum and f″(x₀)<0 for a local maximum. On a closed interval, compare values at critical points and endpoints for absolute extrema.
Worked example
Find intervals of increase/decrease and local extrema for f(x)=x³−3x²+4. f′(x)=3x²−6x=3x(x−2). Sign chart: positive on (−∞,0), negative on (0,2), positive on (2,∞). So f increases on (−∞,0) and (2,∞), decreases on (0,2). At x=0, f changes from increasing to decreasing: local maximum f(0)=4. At x=2, f changes from decreasing to increasing: local minimum f(2)=0.
Traps
A zero derivative does not guarantee an extremum; check sign change. Do not forget endpoints when asked for absolute maximum or minimum. The normal is perpendicular to the tangent, so its slope is negative reciprocal. In rate problems, differentiate with respect to time and substitute the instantaneous values, not average rates.