Download Udemy – Calculus 1, part 2 of 2: Derivatives with applications 2024-3

S1. Introduction of the course: You will get to know the content of this course and the importance of differential calculus. The purpose of this section is not to teach you all the details (that comes later in the course) but to show you the big picture.

S2. Definition of derivative with some examples and examples: You will learn: Formal definition of derivatives and differentiability. terminology and notation; Geometric interpretation of the derivative at a point. Tangent lines and their equations. How to calculate some derivatives directly from the definition and see the result along with the graph of the function in the coordinate system. Continuity vs. Differentiation; higher order derivatives; Differentials and their geometric interpretation. linearization

S3. “Deriving Derivatives of Elementary Functions” You will learn: How to derive formulas for derivatives of basic elementary functions: constant function, Mooney monomials, roots, trigonometric and inverse functions, exponential functions, logarithmic functions, and some power functions (more in the next section) How to prove and apply the sum law, scaling law, product law, and quotient law to derivatives, and how to use these laws to distinguish a large number of new elementary functions formed from basis functions. Differentiation of continuous piecewise functions is defined with the help of elementary functions.

S4. The Chain Law and Related Rates: You will learn: How to calculate the derivatives of complex functions using the Chain Law. Some pictures and proof of the chain law. Derivatives Derivatives formulas are a more general type of power functions, and exponential functions with a different base than e. How to solve some types of related rate problems (problems that can be solved with the help of the chain law).

S5. Derivatives of Inverse Functions: You will learn to: Formulate the derivative of an inverse function to a separable inverse function defined on an interval (with a very good geometric/trigonometric intuition behind it). We revisit some of the formulas derived earlier in the course and show how they can be motivated by the new theorem, but you will also see other examples of applications of this theorem.

S6. Mean value theorems and other important theorems: You will learn: various theorems that play an important role for later applications: mean value theorems (Lagrange, Cauchy), Darbox property, Rolle’s theorem, Fermat’s theorem. You will learn formulations, proofs, intuitive/geometric interpretations, examples of applications, the importance of various assumptions. You will learn some new terms such as CP (critical point, aka stationary point) and singular point. The definitions of maximum/relative minimum and global/absolute maximum/minimum will be repeated from Calculus 1 so that we can use them in the calculus context (they will be discussed more practically in Sections 7 and 17, and 18).

S7. Applications: Uniformity and Optimization: You will learn: How to apply the results of the previous section in more practical settings such as checking the uniformity of differentiable functions and optimizing (mainly continuous) functions. First derivative test and second derivative test for classification of CP (critical points) of distinct functions.

S8. Convexity and Second Derivatives: You will learn: How to use the second derivative to determine whether a function is convex on a concave-convex interval. Milestones and how to see them in the graph of functions. The concept of convexity is a general one, but here we apply it only to doubly differentiable functions.

S9. l’Hôpital’s Law with Applications: You will learn: Use l’Hôpital’s Law to calculate the limits of indeterminate shapes. You get a very detailed proof in the article attached to the first video in this section.

S10. Higher order derivatives and an introduction to the Taylor formula: You will learn: about the classes of real-valued functions of a real variable: C^0, C^1, … , C^∞ and some prominent members of these classes. Importance of Taylor/Maclarin polynomials and their form for exponential function, for sine and for cosine. You will only get a glimpse of these topics, as they are usually part of Account 2.

S11. Implicit Differentiation: You will learn: How to find the derivative y'(x) of an implicit relation F(x,y)=0 by combining different rules for differentiation. You will find examples of curves described by implicit relations, but the study of them is not included in this course (usually studied in “algebraic geometry”, “differential geometry”, or “geometry and topology”; the subject is also Hadi in “Calculus 3 (Multivariable Calculus), Part 1 of 2”: The Implicit Function Theorem).

S12. Logarithmic Differentiation: You will learn: how to perform logarithmic differentiation and in what cases it is practical to apply it.

S13. Briefly about partial derivatives: You will learn: How to calculate partial derivatives for multivariate functions (just an introduction).

S14. Briefly about antiderivatives: You will learn: about the extraordinary application of integrals and about basic integration techniques.

S15. A very short introduction to the subject of ODEs: You will learn: A few very basic things about ordinary differential equations.

S16. More advanced concepts are built on the concept of derivative: You will learn: about some more advanced concepts based on the concept of derivative: partial derivative, gradient, Jacobian, Hessian, derivative of vector-valued functions, divergence, rotation (skew).

S17. Problem Solving: Optimization: You will learn: How to solve optimization problems (practice to Section 7).

S18. Problem Solving: Graphing Functions: You will learn: How to construct a table of changes (sign) of a function and its derivatives. You get a lot of practice graphing functions (the topic is partially covered in “Calculus 1, Part 1 of 2: Limits and Continuity” and is completed in Parts 6-8 of this course).

S19. Extras: You will learn: about all the courses we offer. You’ll also get a glimpse of our plans for future courses, with approximate (very hypothetical!) release dates.

Be sure to check with your professor what parts of the course you need for your final exam. Such cases vary from country to country, university to university, and can even vary from year to year within the same university.

Installation guide

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