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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:

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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Bibliography (further reading)

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983

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