Calculus
Core ideas
Differentiation from first principles, the power, product, quotient and chain rules, and derivatives of sin, cos, tan, eˣ and ln x. Interpret derivatives as gradients and rates of change. Find equations of tangents and normals, stationary points, and classify them with the second derivative or a sign table. For integration, know the reverse power rule, integrals of sin, cos, eˣ and 1/x, definite integrals as signed area, and the fundamental theorem. Apply calculus to optimisation and kinematics: velocity is the derivative of displacement, acceleration the derivative of velocity.
Traps
- Forgetting the chain rule's inner derivative, e.g. d/dx sin(3x) = 3cos(3x).
- Sign errors with the quotient rule; keep numerator order consistent.
- Ignoring the constant of integration in indefinite integrals.
- Confusing displacement with distance travelled; distance requires splitting at turning points.
- Using degrees in calculus; all trigonometric derivatives assume radians.
- Stationary points need classification, not just location.
Worked example
Find the maximum area of a rectangle with perimeter 20 cm. Let sides be x and 10 − x, area A = x(10 − x) = 10x − x². Then A′ = 10 − 2x = 0 gives x = 5. A″ = −2 < 0, so maximum. Area = 25 cm². Rule of thumb: define variables, write the constraint, express the target as one variable, differentiate, solve, then justify with the second derivative.