## Calculus: Single Variable Part 2 - Differentiation

Start Date: 08/16/2020

 Course Type: Common Course

Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this second part--part two of five--we cover derivatives, differentiation rules, linearization, higher derivatives, optimization, differentials, and differentiation operators.

#### Course Syllabus

Think derivatives mean "slopes"? Not anymore... In this module, we will reconsider what a derivative is and means in terms of the asymptotic (or big-O) notation from the previous chapter. This will give us a new language for describing and understanding rates of change and the rules that govern them.

#### Course Introduction

Calculus: Single Variable Part 2 - Differentiation Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) a clear, dynamic, unified approach; and 4) a clear, unified approach with an emphasis on application. In this second part--part two of five--we cover different types of differential equations. We cover linearization, polynomialization, exponential and logarithmic optimization, and we also introduce optimization over a variety of different situations, including optimization problems in series and polynomial programs, as well as applications of polynomial to sequences of n variables. In this second part--part two of five--we cover different types of differential equations. We cover linearization, polynomialization, exponential and logarithmic optimization, and we also introduce optimization problems in series and polynomial programs, as well as applications of polynomial to sequences of n variables. In this second part--part two of five--we cover different types of differential equations. We

#### Course Tag

Differential (Mathematics) Newton'S Method Linear Approximation Differential Calculus Derivative

#### Related Wiki Topic

Article Example
Multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables, rather than just one.
Multivariable calculus In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus:
Derivative The process of finding a derivative is called differentiation. The reverse process is called "antidifferentiation". The fundamental theorem of calculus states that antidifferentiation is the same as integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.
Differential calculus Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.
Logarithmic differentiation In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function "f",
Notation for differentiation In differential calculus, there is no single uniform notation for differentiation. Instead, several different notations for the derivative of a function or variable have been proposed by different mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the antidifferentiation or indefinite integration) are listed below.
Constant factor rule in differentiation In calculus, the constant factor rule in differentiation allows one to take constants outside a derivative and concentrate on differentiating the function of x itself. This is a part of the linearity of differentiation.
Differentiation rules This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
Function of several real variables Elementary calculus is the calculus of real-valued functions of one real variable, and the principal ideas of differentiation and integration of such functions can be extended to functions of more than one real variable; this extension is multivariable calculus.
Sum rule in differentiation In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. This is a part of the linearity of differentiation. The sum rule in integration follows from it. The rule itself is a direct consequence of differentiation from first principles.
Calculus The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the fundamental theorem of calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.
Calculus Leibniz and Newton are usually both credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known.
Itô calculus Malliavin calculus provides a theory of differentiation for random variables defined over Wiener space, including an integration by parts formula .
Operational calculus The key element of the operational calculus is to consider differentiation as an operator "p" = ⁄ acting on functions.
Vector calculus identities The gradient of the product of two scalar fields formula_5 and formula_32 follows the same form as the product rule in single variable calculus.
Calculus Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". "Calculus" (plural "calculi") is also used for naming some methods of calculation or theories of computation, such as propositional calculus, calculus of variations, lambda calculus, and process calculus.
Variable (mathematics) The concept of a "variable" is also fundamental in calculus.
Lambda calculus Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any single-taped Turing machine and was first introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics.
Notation for differentiation When more specific types of differentiation are necessary, such as in multivariate calculus or tensor analysis, other notations are common.
AP Calculus The material includes the study and application of differentiation and integration, and graphical analysis including limits, asymptotes, and continuity. An AP Calculus AB course is typically equivalent to one semester of college calculus.