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4: Differentiation

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    In this chapter, we discuss basic properties of the derivative of a function and several major theorems, including the Mean Value Theorem and l'Hôpital's Rule.

    • 4.1: Definition and Basic Properties of the Derivative
      This page explores differentiability of functions on open subsets of \(\mathbb{R}\), defining differentiability at a point and emphasizing continuity. It covers differentiation rules and the chain rule, alongside examples, including \(f(x) = x^2\) and composite functions. Notation for continuously differentiable functions \(C^{1}(I)\) and higher-order derivatives \(C^{n}(I)\) is introduced.
    • 4.2: THE MEAN VALUE THEOREM
      This page covers the Mean Value Theorem and its connection to Rolle's Theorem and Fermat's Rule, emphasizing their roles in calculus and optimization. It discusses applications of the Mean Value Theorem to establish inequalities and introduces Cauchy's Theorem, focusing on derivatives' properties. The page concludes with exercises that challenge students to apply these concepts in proving equations and exploring limits related to differentiable functions.
    • 4.3: SOME APPLICATIONS OF THE MEAN VALUE THEOREM
      This page discusses the connection between a function's derivative and its characteristics like constancy and monotonicity, asserting that a differentiable function with a zero derivative is constant. It also details criteria for identifying functions as strictly increasing or decreasing, and mentions the Inverse Function Theorem.
    • 4.4: L'Hopital's Rule
      This page covers L'Hospital's Rule for computing limits involving derivatives, highlighting the necessary conditions for its application. It includes examples demonstrating both successful and unsuccessful uses of the rule, such as when differentiability does not ensure continuity.
    • 4.5: Taylor's Theorem
      This page covers Taylor's Theorem, which approximates differentiable functions with polynomials if they have continuous derivatives. It explains how to express a function \(f(x)\) as a Taylor polynomial plus an error term involving the \((n+1)\)-th derivative. The page includes examples, such as approximating \(\sin x\) and analyzing local extrema of functions like \(f(x) = x^2 \cos x\).
    • 4.6: CONVEX FUNCTIONS AND DERIVATIVES
      This page elaborates on convex functions, defining them based on specific inequalities and providing examples like \(f(x) = x\) and \(f(x) = x^2\). It discusses crucial theorems regarding their properties, particularly the conditions for absolute minima and the connection between derivatives and convexity.
    • 4.7: NONDIFFERENTIABLE CONVEX FUNCTIONS AND SUBDIFFERENTIALS
      This page provides an overview of subderivatives and subdifferentials for convex functions, essential for non-differentiable optimization problems. It establishes that a function is differentiable at a point if its subderivative set is a singleton and discusses operations preserving convexity. It further explores absolute minima and conditions for Lipschitz continuity related to bounded subdifferentials.

    This page titled 4: Differentiation is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Lafferriere, Lafferriere, and Nguyen (PDXOpen: Open Educational Resources) via source content that was edited to the style and standards of the LibreTexts platform.