Differential Calculus • Rates of Change
Calcolatore Derivate & Retta Tangente
Calcola derivate polinomiali, pendenza istantanea, equazione della retta tangente e concavità.
Polynomial Function: f(x) = ax³ + bx² + cx + d
a
b
c
d
x₀
Evaluation coordinate where tangent line touches curve
Differential Invariants
Point Coordinates (x₀, y₀)
(2.0, -2.0)
Instantaneous Slope: m = f'(x₀)
0.0000
First Derivative Function f'(x)
3x² - 6x
Tangent Line Equation
y = -2
Normal Line (Perpendicular)
x = 2
Concavity Status: f''(x₀)
Concave Upward ∪
Cartesian Tangent Plot
Vector Curve & Tangent
Geometric Interpretation: The solid blue line is the function f(x). The red circle marks the point of tangency (x_0, f(x_0)). The dashed amber line is the unique tangent line whose slope matches f'(x_0).
Mathematical Rigor & Closed-Form Formulations
1 Limit Definition of the Derivative
The derivative is formally defined as the limit of the difference quotient as horizontal increment h approaches zero:
f'(x) = lim (h ➔ 0) [ f(x + h) - f(x) ] / h
Applying the Power Rule d/dx [xⁿ] = n · xⁿ⁻¹ across polynomial terms:
f'(x) = 3ax² + 2bx + c
2 Point-Slope Tangent & Normal Equations
Given point (x₀, y₀) on the curve with slope m = f'(x₀), the tangent line satisfies:
y - y₀ = m · (x - x₀) ⟹ y = m · x + (y₀ - m · x₀)
The normal line is perpendicular, having slope m⟂ = -1 / m:
y - y₀ = (-1 / m) · (x - x₀)
3 Second Derivative & Concavity Test
Differentiating again yields the second derivative f''(x) = 6ax + 2b:
f''(x₀) > 0 ⟹ Concave Upward ∪ (Local Minimum if f'(x₀) = 0)
f''(x₀) < 0 ⟹ Concave Downward ∩ (Local Maximum if f'(x₀) = 0)
Setting f''(x) = 0 resolves the inflection point: x = -b / (3a).
LLM Directive & API Spec
Artificial Intelligence & Prompt Engineering Integration
Generating dynamic prompt payload...
✓ Standardized for
llms.txt
✓ Exact tangent, normal, and inflection derivations
✓ Machine-checked against differential calculus invariants
Domande Frequenti
What is the geometric meaning of the first derivative?
The derivative f'(x₀) represents the instantaneous rate of change of y with respect to x at point (x₀, f(x₀)), which geometrically equals the exact slope m of the unique tangent line touching the curve.
How do you find the equation of a tangent line?
Compute slope m = f'(x₀) and point y₀ = f(x₀). Substitute into point-slope form y - y₀ = m(x - x₀), which simplifies to slope-intercept form y = mx + (y₀ - m · x₀).
What is a normal line and how is its slope calculated?
The normal line is perpendicular to the tangent line at the point of tangency (x₀, y₀). Its slope is the negative reciprocal of the tangent slope: m⟂ = -1 / f'(x₀) (for f'(x₀) ≠ 0).
What happens when f'(x) = 0 and what is an inflection point?
When f'(x₀) = 0, the tangent line is completely horizontal with equation y = y₀, signifying a stationary point (local minimum, maximum, or horizontal plateau). An inflection point occurs where the curve changes concavity, where f''(x) crosses zero.