Cambridge IGCSE Additional Mathematics Notes 14: Calculus

Cambridge IGCSE Additional Mathematics 0606 Topic 14 develops differentiation and integration across algebraic, trigonometric, exponential and logarithmic functions. It applies calculus to gradients, stationary points, connected rates, approximations, optimisation, plane areas and straight-line kinematics. No calculus formulas are supplied, first-principles differentiation is not required, and points of inflexion are excluded.

1. Derived functions and limits

The derivative measures instantaneous rate of change and graph gradient. Cambridge expects only an informal limit idea: as an increment δx approaches zero, the secant gradient approaches the tangent gradient. Differentiation from first principles is explicitly not required.

Required notation includes f'(x), f''(x), dy/dx, d²y/dx², δx and δx approaching zero. Interpret first and second derivatives in context rather than treating them only as symbols.

2. Standard derivatives

Candidates know derivatives of xⁿ for any rational n, sin x, cos x, tan x, eˣ and ln x. Trigonometric differentiation always uses radians.

The power rule gives d/dx(xⁿ) = nx^(n - 1). Constant multiples and sums differentiate term by term. Standard trigonometric results are:

• derivative of sin x is cos x;
• derivative of cos x is -sin x;
• derivative of tan x is sec²x.

The derivative of eˣ is eˣ, and derivative of ln x is 1/x for x > 0.

3. Chain, product and quotient rules

For a composite f(g(x)), multiply the outer derivative by the inner derivative. Keep the inner expression visible until the chain factor is included.

For product uv, derivative is u'v + uv'. For quotient u/v, derivative is (u'v - uv')/v². Bracket the entire numerator and preserve the original denominator restrictions.

Choose algebraic simplification before differentiating when it makes the structure easier, but do not cancel terms illegally across addition.

4. Tangents and normals

At x = a, tangent gradient is f'(a). The normal is perpendicular, so its gradient is the negative reciprocal when the tangent gradient is non-zero and finite.

Find the point (a, f(a)) from the original function, then form the line through that point. A horizontal tangent has a vertical normal; handle this special case directly.

5. Stationary points

A stationary point satisfies f'(x) = 0. Solve for x, find y and classify it. A sign change from positive to negative derivative gives a local maximum; negative to positive gives a local minimum.

Alternatively, if f''(a) < 0 the stationary point is a local maximum, and if f''(a) > 0 it is a local minimum. If f''(a) = 0, this test is inconclusive. Points of inflexion are excluded, so do not invent them as an assessed classification.