Differentiation and its applications
Differentiation is the Ehime entrance domain that links limits, local behaviour, and curve shape. Expect questions that ask for a derivative from first principles, a tangent or normal, intervals of increase and decrease, extrema, or a sketch justified by signs of f' and f''.
Core rules
- Definition: f'(a) = lim_{h→0} (f(a+h)-f(a))/h; differentiability implies continuity, not conversely.
- Product, quotient, and chain rules; know derivatives of powers, sin, cos, tan, e^x, log x.
- Tangent at x=a: y - f(a) = f'(a)(x-a). Normal slope is -1/f'(a) when f'(a) ≠ 0.
- Critical points from f'=0 or undefined, then use sign change or f'' to classify. Check endpoints and domain restrictions.
- Sketch with domain, intercepts, asymptotes, symmetry, f' sign, and concavity from f''.
Worked example
For f(x)=x^3-3x^2+2, f'(x)=3x^2-6x=3x(x-2), so critical points are x=0,2. Since f' is positive outside [0,2] and negative inside, x=0 gives a local maximum f(0)=2, while x=2 gives a local minimum f(2)=-2. Also f''=6x-6, so concavity changes at x=1, an inflection point. The tangent at x=1 uses f(1)=0 and f'(1)=-3: y=-3(x-1), or y=-3x+3.
Traps
Do not assume f'=0 is automatically an extremum; a sign check is required. Keep the original domain, especially for logarithms and square roots. Distinguish tangent from normal, and do not lose the negative sign in a quotient derivative. When sketching, mark the coordinates of extrema, not just their x-values.