Gradients, tangents & turning points

Differentiate → solve f′ = 0 → classify with signs. Exam-style y = x³ − 6x² + 9x: find and classify turning points Full method. dy/dx = 3x² − 12x + 9 = 3(x − 1)(x − 3) dy/dx = 0 at x = 1 and x = 3 → points (1, 4), (3, 0) Gradient + → − at 1 (max); − → + at 3 (min) ✓ Maximum (1, 4); minimum (3, 0). ℹ

Tangent/normal

Tangent: y − y₁ = m(x − x₁) with m = f′(x₁). Normal: m_normal = −1/m_tangent. ✕

Given the DERIVATIVE, you must integrate — not differentiate again

A recurring question style gives you f'(x) (the gradient function) and a point the ORIGINAL curve passes through, then asks you to find f(x). The move is to INTEGRATE f'(x), not differentiate it — differentiating again gives f''(x), the wrong direction entirely. Substitute the given point into the integrated function to solve for +c.

Practice for this standard covers the whole topic — flashcards and quiz.