Calculus 1, part 2 of 2: Derivatives with applications

• How to solve problems concerning derivatives of real-valued functions of 1 variable (illustrated with 330 solved problems) and why these methods work.
• Definition of derivatives of real-valued functions of one real variable, with a geometrical interpretation and many illustrations.
• Write equations of tangent lines to graphs of functions.
• Derive the formulas for the derivatives of basic elementary functions.
• Prove, apply, and illustrate the formulas for computing derivatives: the Sum Rule, the Product Rule, the Scaling Rule, the Quotient and Reciprocal Rule.
• Prove and apply the Chain Rule; recognise the situations in which this rule should be applied and draw diagrams helping in the computations.
• Use the Chain Rule in problem solving with related rates.
• Use derivatives for solving optimisation problems.
• Understand the connection between the signs of derivatives and the monotonicity of functions; apply first- and second-derivative tests.
• Understand the connection between the second derivative and the local shape of graphs (convexity, concavity, inflection points).
• Determine and classify stationary (critical) points for differentiable functions.
• Use derivatives as help in plotting real-valued functions of one real variable.
• Main theorems of Differential Calculus: Fermat's Theorem, Mean Value Theorems (Lagrange, Cauchy), Rolle's Theorem, and Darboux Property.
• Formulate, prove, illustrate with examples, apply, and explain the importance of the assumptions in main theorems of Differential Calculus.
• Formulate and prove l'Hospital's rule; apply it for computing limits of indeterminate forms; algebraical tricks to adapt the rule for various situations.
• Higher order derivatives; an intro to Taylor / Maclaurin polynomials and their applications for approximations and for limits (more in Calculus 2).
• Classes of functions: C^0, C^1, ... , C^∞; connections between these classes, and examples of their members.
• Implicit differentiation with some illustrations showing horizontal and vertical tangent lines to implicit curves.
• Logarithmic differentiation: when and how to use it.
• A sneak peek into some future applications of derivatives.

Calculus 1, part 2 of 2: Derivatives with applications

Single variable calculus

S1. Introduction to the course

You will learn: about the content of this course and about importance of Differential Calculus. The purpose of this section is not to teach you all the details (this comes later in the course) but to show you the big picture.

S2. Definition of the derivative, with some examples and illustrations

You will learn: the formal definition of derivatives and differentiability; terminology and notation; geometrical interpretation of derivative at a point; tangent lines and their equations; how to compute some derivatives directly from the definition and see the result it gives together with the graph of the function in the coordinate system; continuity versus differentiability; higher order derivatives; differentials and their geometrical interpretation; linearization.

S3. Deriving the derivatives of elementary functions

You will learn: how to derive the formulas for derivatives of basic elementary functions: the constant function, monic monomials, roots, trigonometric and inverse trigonometric functions, exponential functions, logarithmic functions, and some power functions (more to come in the next section); how to prove and apply the Sum Rule, the Scaling Rule, the Product Rule, and the Quotient Rule for derivatives, and how to use these rules for differentiating plenty of new elementary functions formed from the basic ones; differentiability of continuous piecewise functions defined with help of the elementary ones.

S4. The Chain Rule and related rates

You will learn: how to compute derivatives of composite functions using the Chain Rule; some illustrations and a proof of the Chain Rule; derivations of the formulas for the derivatives of a more general variant of power functions, and of exponential functions with the basis different than e; how to solve some types of problems concerning related rates (the ones that can be solved with help of the Chain Rule).

S5. Derivatives of inverse functions

You will learn: the formula for the derivative of an inverse function to a differentiable invertible function defined on an interval (with a very nice geometrical/trigonometrical intuition behind it); we will revisit some formulas that have been derived earlier in the course and we will show how they can be motivated with help of the new theorem, but you will also see some other examples of application of this theorem.

S6. Mean value theorems and other important theorems

You will learn: various theorems that play an important role for further applications: Mean Value Theorems (Lagrange, Cauchy), Darboux property, Rolle's Theorem, Fermat's Theorem; you will learn their formulations, proofs, intuitive/geometrical interpretations, examples of applications, importance of various assumptions; you will learn some new terms like CP (critical point, a.k.a. stationary point) and singular point; the definitions of local/relative maximum/minimum and global/absolute maximum/minimum will be repeated from Precalculus 1, so that we can use them in the context of Calculus (they will be discussed in a more practical way in Sections 7, 17, and 18).

S7. Applications: monotonicity and optimisation

You will learn: how to apply the results from the previous section in more practical settings like examining monotonicity of differentiable functions and optimising (mainly continuous) functions; The First Derivative Test and The Second Derivative Test for classifications of CP (critical points) of differentiable functions.

S8. Convexity and second derivatives

You will learn: how to determine with help of the second derivative whether a function is concave of convex on an interval; inflection points and how they look on graphs of functions; the concept of convexity is a general concept, but here we will only apply it to twice differentiable functions.

S9. l'Hôpital's rule with applications

You will learn: use l'Hôpital's rule for computing the limits of indeterminate forms; you get a very detailed proof in an article attached to the first video in this section.

S10. Higher order derivatives and an intro to Taylor's formula

You will learn: about classes of real-valued functions of a single real variable: C^0, C^1, ... , C^∞ and some prominent members of these classes; the importance of Taylor/Maclaurin polynomials and their shape for the exponential function, for the sine and for the cosine; you only get a glimpse into these topics, as they are usually a part of Calculus 2.

S11. Implicit differentiation

You will learn: how to find the derivative y'(x) from an implicit relation F(x,y)=0 by combining various rules for differentiation; you will get some examples of curves described by implicit relations, but their study is not included in this course (it is usually studied in "Algebraic Geometry", "Differential Geometry" or "Geometry and Topology"; the topic is also partially covered in "Calculus 3 (Multivariable Calculus), part 1 of 2": Implicit Function Theorem).

S12. Logarithmic differentiation

You will learn: how to perform logarithmic differentiation and in what type of cases it is practical to apply.

S13. Very briefly about partial derivatives

You will learn: how to compute partial derivatives to multivariable functions (just an introduction).

S14. Very briefly about antiderivatives

You will learn: about the wonderful applicability of integrals and about the main integration techniques.

S15. A very brief introduction to the topic of ODE

You will learn: some very basic stuff about ordinary differential equations.

S16. More advanced concepts built upon 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 (curl).

S17. Problem solving: optimisation

You will learn: how to solve optimisation problems (practice to Section 7).

S18. Problem solving: plotting functions

You will learn: how to make the table of (sign) variations for the function and its derivatives; you get a lot of practice in plotting functions (topic covered partly in "Calculus 1, part 1 of 2: Limits and continuity", and completed in Sections 6-8 of the present course).

S19. Extras

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

Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.

A detailed description of the content of the course, with all the 245 videos and their titles, and with the texts of all the 330 problems solved during this course, is presented in the resource file

“001 List_of_all_Videos_and_Problems_Calculus_1_p2.pdf”

under video 1 ("Introduction to the course"). This content is also presented in video 1.