Calculus: Further Differentiation, Integration and Applications
What you must understand
This domain covers techniques and their applications: implicit and parametric differentiation, logarithmic differentiation, higher derivatives, integration by parts, substitution, partial fractions, trigonometric integrals, and the calculus of inverse trigonometric functions. Applications include curve sketching, maxima and minima, rates of change, volumes of solids of revolution, arc length, and simple differential equations solved by separation of variables.
Traps
- Chain rule omissions in implicit differentiation: every (y) term differentiated with respect to (x) picks up a factor (\frac{dy}{dx}).
- Parametric second derivative: (\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}), not (\frac{d^2y/dt^2}{d^2x/dt^2}).
- Integration by parts sign: (\int u,dv = uv - \int v,du); a dropped minus is fatal.
- Definite integral limits after substitution: change the limits, or convert back before evaluating.
- Volume axis: rotating about the (x)-axis uses (\pi\int y^2,dx); about the (y)-axis you need (x) as a function of (y) or shells.
- Domain restrictions for inverse trig derivatives, e.g. (\frac{d}{dx}\sin^{-1}x = \frac{1}{\sqrt{1-x^2}}) only for (|x|<1).
Rule of thumb
Before integrating, classify the integrand: a product suggests parts, a composite suggests substitution, a rational function suggests partial fractions.
Worked example
Evaluate (\int x e^{x},dx). Let (u = x), (dv = e^x dx), so (du = dx), (v = e^x). Then [ \int x e^x,dx = x e^x - \int e^x,dx = e^x(x-1) + C. ]