Inverse Functions and Angles Between Curves

This final page focuses on the derivatives of inverse trigonometric functions and introduces the concept of angles between curves.

The page provides formulas for the derivatives of inverse trigonometric functions:

Formula:

• D[arcsin x] = 1 / √(1-x^2)
• D[arccos x] = -1 / √(1-x^2)
• D[arctan x] = 1 / (1+x^2)
• D[arccot x] = -1 / (1+x^2)

These formulas are crucial for understanding derivata funzione inversa esempi and derivata della funzione inversa esercizi svolti.

The page concludes with a discussion on finding the angle between two curves at their intersection point. It provides the formula:

Formula: tan θ = |m₁ - m₂| / (1 + m₁m₂)

Where m₁ and m₂ are the slopes of the tangent lines to the curves at the intersection point.

Highlight: The concept of angles between curves has important applications in geometry and physics, particularly in understanding the behavior of intersecting paths or trajectories.

Tangent Lines and Derivatives

This page introduces the concept of tangent lines to a function and their relation to derivatives. It explains how to find the equation of a tangent line using limits and the incremental ratio.

The page begins by discussing the classical problem of determining the tangent line to a curve at a given point. It introduces the concept of a secant line and explains how the tangent line is the limit position of the secant line as a second point approaches the point of tangency.

Definition: The tangent line at point P is the limit position of the secant line obtained by making point Q approach point P.

The page then introduces the formula for the slope of the tangent line:

Formula: m(tangent at P) = lim(h→0) [f(x₀+h) - f(x₀)] / h

This formula is crucial for understanding the retta tangente in un punto formula and the coefficiente angolare retta tangente in un punto.

Highlight: The concept of limits is fundamental to understanding derivatives and tangent lines in calculus.

Incremental Ratio and Derivability

This page delves deeper into the concept of the incremental ratio and introduces the formal definition of derivability. It explains the geometric interpretation of the incremental ratio and its relation to the slope of the secant line.

The page defines the incremental ratio as:

Formula: [f(x₀+h) - f(x₀)] / h

Where h represents the increment of the independent variable.

Vocabulary: The incremental ratio is also known as the rapporto incrementale in Italian.

The page then provides the formal definition of derivability:

Definition: A function f(x) is derivable at a point x₀ if:

1. It is defined in a neighborhood of x₀
2. The limit of the incremental ratio exists and is finite as h approaches 0

The page also introduces an important theorem:

Theorem: If f(x) is derivable at x₀, then it is continuous at x₀.

This theorem highlights the relationship between derivability and continuity, which is crucial for understanding the definizione di derivata.

Example: The geometric representation of the incremental ratio is the slope of the secant line passing through points P and Q on the function's graph.