Derivation and Function Analysis
I. Derivative Fundamentals
Formal Definition
f'(a) =
lim
h→0
f(a+h) − f(a)
h
Geometric interpretation: f'(a) represents the slope of the tangent line to the curve at x = a.
II. Derivation Rules
Basic Derivatives
| Function | Derivative |
|---|---|
| xn (n ∈ ℕ*) | nxn-1 |
| 1 x | − 1 x2 |
| sin x | cos x |
Operation Rules
Sum Rule:
(u + v)’ = u’ + v’
Product Rule:
(uv)’ = u’v + uv’
Quotient Rule:
⎛
u
v
⎞′ =
u’v − uv’
v2
III. Applications
Function Behavior
For a differentiable function f on interval I:
- If ∀x ∈ I, f'(x) ≥ 0 ⇒ f increasing on I
- If ∀x ∈ I, f'(x) ≤ 0 ⇒ f decreasing on I
- If ∀x ∈ I, f'(x) = 0 ⇒ f constant on I
f'(x) > 0 ⇒ f strictly increasing
Local Extrema
Necessary condition:
If f has a local extremum at a and is differentiable at a, then f'(a) = 0.
Practical method:
- Solve f'(x) = 0
- Study the sign of f’ around critical points
- Determine the nature of critical points
IV. Complete Function Analysis
Methodology
- Domain of definition
- Limits at boundaries and asymptotes
- First derivative f'(x)
- Variation table
- Graphical representation
Example: f(x) = 2x3 – 3x2 – 12x + 5
1. Domain: Df = ℝ
2. Derivative:
f'(x) = 6x2 − 6x − 12 = 6(x − 2)(x + 1)
3. Variation table:
| x | -∞ | -1 | 2 | +∞ |
| f'(x) | + | 0 | – | + |
| f(x) | ↗ | 12 | ↘ | ↗ |
4. Extrema:
- Local maximum at (-1,12)
- Local minimum at (2,-15)
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