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Section: 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).

Table of Contents

1. Definition: Difference Quotient
2. Definition: Derivative, Differentiable Functions
3. Definition: Higher-Order Derivatives
4. Definition: Local Extremum
5. Theorem: Rolle's Theorem
6. Proposition: Differentiable Functions and Tangent-Linear Approximation
7. Proposition: Characterization of Monotonic Functions via Derivatives
8. Theorem: Darboux's Theorem
9. Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule
10. Proposition: Chain Rule

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983