Differential Calculus
Scholarship Calculus uses Differential calculus for multi-step analysis: differentiability, chain, product and quotient rules, implicit and parametric differentiation, related rates, optimisation, and curve behaviour. Expect justification of signs and existence, not just a derivative.
Core rules and concepts
- Differentiability implies continuity; continuity does not imply differentiability, as with |x| at 0.
- Chain rule: d/dx f(g(x)) = f'(g(x))g'(x). For parametric x(t), y(t), dy/dx = (dy/dt)/(dx/dt), provided dx/dt≠0.
- Implicit differentiation: differentiate both sides with respect to x, treating y as y(x), then collect dy/dx.
- Stationary points require f'=0 or f' undefined; classify with sign changes or f'' where valid.
- Optimisation on a closed interval must compare stationary values with endpoint values.
Worked example
For f(x)=x^2 e^{-x}, find and classify stationary points. Product and chain rules: f'(x)=2x e^{-x} + x^2(-e^{-x}) = x(2-x)e^{-x}. Since e^{-x}>0, f'=0 at x=0 and x=2. Differentiate again: f''(x)= e^{-x}(x^2-4x+2). At x=0, f''=2>0, so local minimum. At x=2, f''=-2e^{-2}<0, so local maximum. This also shows the graph decreases after x=2.
Traps
Dropping the inner derivative; using f''=0 as a classification when it may be inconclusive; ignoring points where f' is undefined; treating dy/dx as a fraction in related rates without tracking variables; forgetting endpoints in closed-interval optimisation.