Calculus: Differentiation, Integration and Applications
Core ideas
Differentiation from first principles defines the derivative as a limit of a difference quotient; in practice you apply the power, product, quotient and chain rules, and differentiate exponentials, logarithms and trigonometric functions. The derivative gives gradient, so it drives tangents and normals, stationary points, increasing/decreasing intervals and rates of change.
Integration reverses differentiation: indefinite integrals need + C, and definite integrals give signed area. Area between curves is ∫(upper − lower) dx, found by locating intersections first. The trapezoidal rule approximates definite integrals and is examined with a table of values.
Traps
- The chain rule is the most frequent omission: d/dx of sin(3x) is 3cos(3x), not cos(3x).
- A stationary point is not automatically a maximum or minimum - justify with the second derivative or a sign table.
- Definite integrals below the axis are negative; split the interval when the curve crosses the x-axis if total area is required.
- Differentiate before substituting; substituting first destroys the variable.
- Normal gradient is the negative reciprocal of the tangent gradient.
Worked example
Find the stationary points of f(x) = x³ − 3x² + 4 and classify them. f′(x) = 3x² − 6x = 3x(x − 2), so x = 0 or x = 2. f″(x) = 6x − 6, giving f″(0) = −6 (local maximum, f(0) = 4) and f″(2) = 6 (local minimum, f(2) = 0).
Rule of thumb
For any optimisation or curve-sketching question, write the derivative, solve f′(x) = 0, then test - and always answer the question actually asked (coordinates, not just x-values).