## Differentiable Functions

Many properties of real-valued functions are reflected by their derivatives. Thus, the study of differentiation of real-valued functions helps to better understand the following:

- The occurrence of their local minima or maxima (see example),
- Their monotonic behavior (see example),
- Their convexity (see example).

Moreover, the growth and fluctuations of the functions can be estimated by boundaries of their derivatives (see example).

| | | | Contributors: *bookofproofs* | References: [581]

## 1.**Definition**: Difference Quotient

## 2.**Definition**: Derivative, Differentiable Functions

## 3.**Definition**: Higher-Order Derivatives

## 4.**Definition**: Local Extremum

## 5.**Theorem**: Rolle's Theorem

## 6.**Definition**: Continuously Differentiable Functions

## 7.**Proposition**: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule

## 8.**Proposition**: Differentiable Functions and Tangent-Linear Approximation

## 9.**Proposition**: Chain Rule

## 10.**Proposition**: Characterization of Monotonic Functions via Derivatives

## 11.**Theorem**: Darboux's Theorem

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[581] **Forster Otto**: “Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen”, Vieweg Studium, 1983

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