Calculus 1, part 2 of 2: Derivatives with applications from Udemy

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.

What's inside

Learning objectives

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.
Show more
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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.
Show more
Show less

Syllabus

Introduction to the course

Good news first

Rates of change, slopes, and tangent lines

Derivative at a point and derivative as a function that shows variable slopes

Why derivatives are important

Differential equations: find all the functions that change in a certain way

Elementary functions and their derivatives: more and less intuitive rules

Advanced topics in the Precalculus series

Definition of the derivative, with some examples and illustrations

Terminology and notation

Where to find Precalculus stuff for repetition: straight lines, rates of change

In what kind of points we are going to consider derivatives

Definition of the derivative at a point, differentiability of functions

How to find equations for tangent lines? Two methods

Derivatives of linear functions, Exercise 1

Derivatives of quadratic functions, Exercise 2

Derivatives of quadratic functions, Exercise 3

Derivative of a cubic polynomial, Exercise 4

Derivative of the square root function, Exercise 5

Another (equivalent) way of defining derivatives, Exercise 6

A function that is not differentiable at some point, Exercise 7

Absolute values and cusps: a generalisation of Exercise 7

Yet another way of defining differentiability at a point

Each differentiable function is continuous, but is the converse true?

Optional: Proof of the part C1 from the theorem in Video 21

Optional: Proof of the part C2.1 from the theorem in Video 21

Is the absolute value always a bad news for global differentiability? Problem 1

Derivatives of piecewise functions, Problem 2

Recognising derivatives, Problem 3

Recognising derivatives, Problem 4

Recognising derivatives, Problem 5

Computing derivatives from the definition, Problem 6

One of my favourite problems, Problem 7

Higher order derivatives, definition and notation

Geometric interpretation of differentials

What is linearization and why it is good for you

Linearization works locally, Problem 8

Deriving the derivatives of elementary functions

Our plan

The derivative of monic monomials (power functions 1), method 1

The derivative of monic monomials (power functions 1), method 2

The derivative of roots (power functions 2), method 1

The derivative of power functions 3, method 1

The derivative of sine, method 1

The derivative of cosine, method 1

The derivative of sine and cosine, method 2

The derivative of sine inverse, method 1

The derivative of cosine inverse

The derivative of the exponential function

The derivative of the natural logarithm, method 1

The derivative of logarithms with any base

Rules of differentiation: the main theorem

An illustration for the Sum Rule

Some illustrations for the Product Rule

Three ways of writing a proof of the Sum Rule

Three ways of writing a proof of the Scaling Rule

Linearity of the differential operator, and its consequences

A proof of the Product Rule

Derivatives of polynomials are polynomials

A proof of the Quotient (and Reciprocal) Rule

The derivative of power functions 3, method 2

Derivatives of rational functions are rational functions

A generalization of the Product Rule

The derivative of tangent

The derivative of arctangent, method 1

Some practice in differentiation, Exercise 1

Some practice in differentiation, Exercise 2

Some practice in differentiation, Exercise 3

Some practice in differentiation, Exercise 4

Some practice in differentiation, Exercise 5

Some practice in differentiation, Exercise 6

Some practice in differentiation, Exercise 7

Some practice in differentiation, Exercise 8

Some practice in differentiation, Exercise 9

Some practice in differentiation, Exercise 10

Differentiability of piecewise functions, Exercise 11

The derivative of the sine of a scaled argument

Differentiability of piecewise functions, Exercise 12

Multiple zeros of polynomials and the round shapes of the graphs

A really cool problem about polynomials, Problem 1

The one with a picture, Problem 2

Finding the tangent line, Problem 3

Where to find more exercises for practice; some hints and tricks

The Chain Rule and related rates

About this section; some reading recommendations

Repetition from Precalculus 1: compositions of functions

Transformations of graphs that involve scalings of the argument

The Chain Rule: the theorem, an example, and a proof

A generalization of The Chain Rule and some related topics

Back to Video 87 in Precalculus 1

Back to Video 88 from Pre1: the order of functions in a composition is important

The derivative of power functions 3, method 3

The derivative of power functions 4

Some useful formulas from Precalculus 4

The derivative of power functions 5

The derivative of exponential functions

Neither exponential nor power functions

Back to some details from Videos 48 and 77

The Chain Rule, an example

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Focuses on calculus, which is foundational for individuals interested in math, science, engineering and related fields

Appropriate for learners with a basic understanding of precalculus and some experience with algebra and trigonometry

Taught by instructors with background in mathematics and experience teaching calculus

Emphasis on solving problems related to derivatives, making this suitable for those who want to strengthen their analytical skills

May require additional resources or a strong background for learners who need more foundational support

Course is part of a series, suggesting a more comprehensive approach to calculus

Career center

Learners who complete Calculus 1, part 2 of 2: Derivatives with applications will develop knowledge and skills that may be useful to these careers:

Data Analyst

Data Analysts examine and interpret data that is widely available today in order to turn it into usable information. Since they spend their days grappling with large datasets, a class focused on Calculus can provide the foundation needed to help you get started. For example, you may be tasked with deciding which statistical method is the most effective in revealing trends in a particular customer base. The material in a class like this one can help you in making the right choice, and will also provide you the background you need to gain insights from the data that your team compiles.

See salaries and explore the career path for Data Analyst

Investment Analyst

Investment Analysts spend their days examining market trends, analyzing financial data, and making recommendations as to what stocks to buy. Taking a class on Calculus may not be at the top of your list but it can be a useful way to strengthen your foundational knowledge of math as it pertains to finance and investing. For example, you may find that understanding the derivatives of different asset classes helps you make smarter recommendations to your clients.

See salaries and explore the career path for Investment Analyst

Operations Research Analyst

Operations Research Analysts use analytical methods to solve complex problems in business and industry. Although everyday tasks will not require Calculus, the strong quantitative background that this course can provide will make you a more competitive candidate in the job market.

See salaries and explore the career path for Operations Research Analyst

Financial Analyst

A Financial Analyst needs to have a background in quantitative methods. Although you may not use the explicit materials from Calculus in this role, a strong understanding of the core concepts can set you apart from the competition. For example, being able to calculate the derivative of a function can be helpful when trying to find the optimal solution to a problem.

See salaries and explore the career path for Financial Analyst

Business Analyst

A Business Analyst role requires one to have a strong quantitative background. When working with this kind of data, it is helpful to have a strong understanding of how mathematical functions work and how to get the most out of quantitative data. In this way, Calculus can be helpful for you in this role even though it is not directly used day to day.

See salaries and explore the career path for Business Analyst

Data Scientist

Data Scientists must have a strong background in mathematics and statistics. Calculus is a core component of this background and can help you become a stronger candidate for these roles.

See salaries and explore the career path for Data Scientist

Economist

Economists apply mathematical and statistical techniques to analyze economic data. Although you probably won't be using Calculus on a day-to-day basis, having proficiency in the subject can help open doors in the field and ensure that you are well-prepared for any eventuality.

See salaries and explore the career path for Economist

Market Research Analyst

Market Research Analysts must be able to use and understand quantitative and qualitative data in order to make informed recommendations on how to best market a product or service. This course can be useful for building a foundation in understanding and using quantitative data.

See salaries and explore the career path for Market Research Analyst

Statistician

Statisticians use mathematical and statistical methods to collect, analyze, interpret, and present data. Calculus is a good way to build a foundation for understanding the math and statistics used everyday in this role.

See salaries and explore the career path for Statistician

Computer Scientist

Computer Scientists need to have a solid foundation in mathematics. Calculus is a great way to begin building this base.

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Software Engineer

Software Engineers need to have a strong foundation in mathematics. Taking Calculus can provide you with a strong foundation for success in the field.

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Machine Learning Engineer

Machine Learning Engineers build and maintain machine learning models. Calculus can provide you with a great foundation in the math that is needed in the field.

See salaries and explore the career path for Machine Learning Engineer

Data Engineer

Data Engineers collect, clean, and analyze data to help businesses make better decisions. Calculus can provide a good foundation in the mathematics needed for success in this role.

See salaries and explore the career path for Data Engineer

Quantitative Analyst

Quantitative Analysts use mathematical and statistical methods to analyze and solve financial problems. Calculus can provide a strong foundation for the math needed in this role.

See salaries and explore the career path for Quantitative Analyst

Actuary

Actuaries use mathematical and statistical skills to assess risk. Since these roles tend to require an advanced degree, taking Calculus now can provide you with a strong foundation and show future employers a dedication to your craft.

See salaries and explore the career path for Actuary