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).

| | | | created: 2014-02-23 13:25:00 | modified: 2017-08-01 18:54:43 | by: bookofproofs | references: [581]

1.Definition: Difference Quotient 

2.Definition: Derivative, Differentiable Functions 

3.Definition: Higher-Order Derivatives 

4.Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule 

5.Definition: Local Extremum 

6.Theorem: Rolle's Theorem 

7.Proposition: Differentiable Functions and Tangent-Linear Approximation 

8.Proposition: Chain Rule 

9.Proposition: Characterization of Monotonic Functions via Derivatives 

10.Theorem: Darboux's Theorem