{"id":699,"date":"2025-12-11T22:17:57","date_gmt":"2025-12-11T22:17:57","guid":{"rendered":"https:\/\/itisallmath.com\/?page_id=699"},"modified":"2026-01-12T21:17:34","modified_gmt":"2026-01-12T21:17:34","slug":"699-2","status":"publish","type":"page","link":"https:\/\/itisallmath.com\/es\/mathematics\/699-2\/","title":{"rendered":"Differential Calculus"},"content":{"rendered":"
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\"\"<\/figure>\n\n\n\n

What are you going to learn?<\/h1><\/div>\n\n\n\n

Evaluate the limit of a function, including one-sided and two-sided, using numerical and algebraic techniques and the properties of limits. Determine whether a function is continuous or discontinuous at a point. Calculate the derivative of an algebraic function using the formal definition of the derivative. Find derivatives of functions using the constant, power, sum, difference, product, quotient, and chain rules, and using implicit differentiation. Determine maxima and minima in optimization problems using the derivative. Sketch the graph of functions using the first and second derivatives. Apply the concept of derivative to solve applied problems.<\/p>\n\n\n\n

Content<\/h2><\/div>\n<\/div><\/div>\n\n\n\n
\n
\n

Chapter 1.\u00a0Basic Concepts of Sets Theory<\/strong><\/h3><\/div>\n\n\n\n
    \n
  • Introduction<\/li>\n\n\n\n
  • Notation<\/li>\n\n\n\n
  • Subsets<\/li>\n\n\n\n
  • Operations<\/li>\n\n\n\n
  • NeIghborhood Spaces and Function Spaces<\/li>\n<\/ul>\n<\/div>\n\n\n\n
    <\/div>\n<\/div><\/div>\n\n\n\n
    \n
    \n

    Chapter 2.\u00a0The Real Numbers System<\/strong><\/strong><\/h3><\/div>\n\n\n\n
      \n
    • Introduction<\/li>\n\n\n\n
    • Integers and Rational Numbers<\/li>\n\n\n\n
    • The Field Axioms<\/li>\n\n\n\n
    • The Order Axioms<\/li>\n<\/ul>\n<\/div>\n\n\n\n
      <\/div>\n<\/div><\/div>\n\n\n\n
      \n
      \n

      Chapter 3.\u00a0Functions<\/strong><\/h3><\/div>\n\n\n\n
        \n
      • Representation of Functions<\/li>\n\n\n\n
      • Mathematical Models<\/li>\n\n\n\n
      • Transformation of Functions<\/li>\n\n\n\n
      • Exponential Functions<\/li>\n\n\n\n
      • Inverse Functions and Logarithms<\/li>\n<\/ul>\n<\/div>\n\n\n\n
        <\/div>\n<\/div><\/div>\n\n\n\n
        \n
        \n

        Chapter 4.\u00a0Limits<\/strong><\/h3><\/div>\n\n\n\n
          \n
        • Tangent and Velocity Problems<\/li>\n\n\n\n
        • The Limit of a Function<\/li>\n\n\n\n
        • Calculation of Limits<\/li>\n\n\n\n
        • The Precise Definition of a Limit<\/li>\n\n\n\n
        • Limits at Infinity<\/li>\n<\/ul>\n<\/div>\n\n\n\n
          <\/div>\n<\/div><\/div>\n\n\n\n
          \n
          \n

          Chapter 5.\u00a0Continuity<\/strong><\/h3><\/div>\n\n\n\n
            \n
          • Definition<\/li>\n\n\n\n
          • Cauchy Characterization of Continuity<\/li>\n\n\n\n
          • Continuity of Elementary Functions<\/li>\n\n\n\n
          • General Properties of Continuous Functions<\/li>\n\n\n\n
          • Continuity of Inverse Functions<\/li>\n<\/ul>\n<\/div>\n\n\n\n
            <\/div>\n<\/div><\/div>\n\n\n\n
            \n
            \n

            Chapter 6.\u00a0Derivatives and the Differentiation Rules<\/strong><\/strong><\/h3><\/div>\n\n\n\n
              \n
            • Derivatives of First Order. Definition<\/li>\n\n\n\n
            • Differentiation of Elementary Functions<\/li>\n\n\n\n
            • Differentiation of Inverse Functions<\/li>\n\n\n\n
            • Differentiation of Composite Functions<\/li>\n<\/ul>\n<\/div>\n\n\n\n
              <\/div>\n<\/div><\/div>\n\n\n\n
              \n
              \n

              Chapter 7.\u00a0Applications of Differentiation<\/strong><\/strong><\/h3><\/div>\n\n\n\n
                \n
              • Extrema of Functions. THe Rolle Theorem<\/li>\n\n\n\n
              • Lagrange and Cauchy Theorems<\/li>\n\n\n\n
              • Geometrical Interpretation of the Derivative Sign<\/li>\n\n\n\n
              • Indeterminate Expressions<\/li>\n\n\n\n
              • Asymptotes<\/li>\n\n\n\n
              • The Concept of Differential<\/li>\n<\/ul>\n<\/div>\n\n\n\n
                <\/div>\n<\/div><\/div>\n\n\n\n
                \n
                \n

                Chapter 8.\u00a0Derivatives of Higher Order<\/strong><\/strong><\/h3><\/div>\n\n\n\n
                  \n
                • Definitions and Examples<\/li>\n\n\n\n
                • Differentials of Higher Order<\/li>\n\n\n\n
                • Taylor’s Formula<\/li>\n\n\n\n
                • Expansions in Power Series<\/li>\n\n\n\n
                • The Concept of Differential of Higher Order<\/li>\n\n\n\n
                • Criterium of Extrema<\/li>\n\n\n\n
                • Geometrical Interpretation of the Second Derivative<\/li>\n<\/ul>\n<\/div>\n\n\n\n
                  <\/div>\n<\/div><\/div>\n\n\n\n

                  Bibliography<\/h2><\/div>\n\n\n\n
                    \n
                  • Thomas, G. et al (2005).\u00a0T<\/em>homas’ Calculus:Early Transcendentals<\/em><\/strong>.<\/em>\u00a011th Edition: Pearson Education, Limited.<\/li>\n\n\n\n
                  • Stewart, J.\u00a0Calculus Early Transcendentals<\/em><\/strong>. 8th edition. Belmont, CA: Brooks Cole.<\/li>\n\n\n\n
                  • Apostol, T. (1969).\u00a0Calculus VOL.<\/em><\/strong>\u00a01<\/em>, 2d Ed. John Wiley and Sons..<\/li>\n\n\n\n
                  • Spivak, M. (2006).\u00a0Calculus<\/em><\/strong>. 4th Ed, Cambridge University Press.<\/li>\n<\/ul>\n\n\n\n

                    Bibliography<\/h2><\/div>\n\n\n\n
                      \n
                    • <\/li>\n<\/ul>","protected":false},"excerpt":{"rendered":"

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