A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Functions
CSCA Mathematics · Lesson 9 of 16Derivatives
Dérivées · المشتقات
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The derivative f′(x) is the rate of change of f, and the slope of its tangent line. It is the tool for finding maximums and minimums.
Key ideas
- Power rule: (xⁿ)′ = n·xⁿ⁻¹. The derivative of a constant is 0.
- (f + g)′ = f′ + g′ and (k·f)′ = k·f′.
- Tangent line at x = a: y = f′(a)(x − a) + f(a).
- f has a local maximum or minimum where f′(x) = 0 and f′ changes sign.
Formulas to know
- (sin x)′ = cos x, (cos x)′ = −sin x
- (eˣ)′ = eˣ, (ln x)′ = 1/x
- Product rule: (fg)′ = f′g + fg′
Worked example
Find the minimum of f(x) = x² − 6x + 1.
- f′(x) = 2x − 6.
- f′(x) = 0 when x = 3.
- f(3) = 9 − 18 + 1 = −8, and f′ changes from negative to positive.
Answer: Minimum value −8, at x = 3
Practice questions
1. f(x) = 4x³. f′(x) = ?
12x² ✓ · 4x² · 12x³ · x⁴
3 · 4x² = 12x².
2. Slope of the tangent to y = x² at x = 3?
3 · 6 ✓ · 9 · 2
y′ = 2x = 6 at x = 3.
3. f(x) = x³ − 3x. Where is f′(x) = 0?
x = 0 · x = ±1 ✓ · x = ±3 · x = 3
f′(x) = 3x² − 3 = 0 gives x² = 1, so x = ±1.