Introduction to Differentiation
IB Math AA — the derivative as a gradient, the power rule, tangent lines and increasing/decreasing functions. Worked example and Paper 1 vs Paper 2 advice.
Differentiation is a core thread of IB Mathematics: Analysis & Approaches (AA), at both SL and HL. The derivative f'(x) (or dy/dx) gives the gradient of the curve at any point.
The power rule
f(x) = xⁿ → f'(x) = n·xⁿ⁻¹, applied term by term.
f(x) = 3x⁴ − 5x² + 2x → f'(x) = 12x³ − 10x + 2
Tangent lines
The gradient of the tangent at x = a is f'(a). Then use y − y₁ = m(x − x₁).
f(x) = x²atx = 3:f'(x) = 2x, som = 6, point(3, 9)→ tangenty = 6x − 9.
Increasing / decreasing
f'(x) > 0→ the function is increasing.f'(x) < 0→ decreasing.f'(x) = 0→ a stationary point (maximum, minimum or inflection).
Worked example
Find the stationary point of f(x) = x² − 4x + 1.
f'(x) = 2x − 4 = 0 → x = 2, and f(2) = −3 → stationary point (2, −3) (a minimum).
IB strategy
Paper 1 is non-calculator — you must differentiate and solve by hand, so practise the power rule until it is automatic. Paper 2 allows a GDC — use it to check stationary points and to graph f'(x). In HL, this foundation extends to the product, quotient and chain rules.
İnteraktif Simülasyon
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Deneme Sınavı
Alıştırma Testi
Kendini dene: anında sonuç ve açıklama.
1. If f(x) = 3x⁴ − 5x² + 2x , then f'(x) =
2. At a stationary point of a curve, the gradient f'(x) is:
3. The stationary point of f(x) = x² − 4x + 1 occurs at x =
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