Calculus 1, Part 2 Of 2: Derivatives With Applications

Differential calculus in one variable: theory and applications for optimisation, approximations, and plotting functions

What you'll learn

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.

Requirements

Precalculus (Basic notions, Polynomials and rational functions, Trigonometry, Exponentials and logarithms)

Calculus 1: Limits and continuity (or equivalent)

You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.

Description

Calculus 1, part 2 of 2: Derivatives with applicationsSingle variable calculusS1. Introduction to the courseYou 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 illustrationsYou 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 functionsYou 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 ratesYou 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 functionsYou 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 theoremsYou 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 optimisationYou 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 derivativesYou 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 applicationsYou 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 formulaYou 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 differentiationYou 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 differentiationYou will learn: how to perform logarithmic differentiation and in what type of cases it is practical to apply.S13. Very briefly about partial derivativesYou will learn: how to compute partial derivatives to multivariable functions (just an introduction).S14. Very briefly about antiderivativesYou will learn: about the wonderful applicability of integrals and about the main integration techniques.S15. A very brief introduction to the topic of ODEYou will learn: some very basic stuff about ordinary differential equations.S16. More advanced concepts built upon the concept of derivativeYou 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: optimisationYou will learn: how to solve optimisation problems (practice to Section 7).S18. Problem solving: plotting functionsYou 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. ExtrasYou 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.

Overview

Section 1: Introduction to the course

Lecture 1 Introduction to the course

Lecture 2 Good news first

Lecture 3 Rates of change, slopes, and tangent lines

Lecture 4 Derivative at a point and derivative as a function that shows variable slopes

Lecture 5 Why derivatives are important

Lecture 6 Differential equations: find all the functions that change in a certain way

Lecture 7 Elementary functions and their derivatives: more and less intuitive rules

Lecture 8 Advanced topics in the Precalculus series

Section 2: Definition of the derivative, with some examples and illustrations

Lecture 9 Terminology and notation

Lecture 10 Where to find Precalculus stuff for repetition: straight lines, rates of change

Lecture 11 In what kind of points we are going to consider derivatives

Lecture 12 Definition of the derivative at a point, differentiability of functions

Lecture 13 How to find equations for tangent lines? Two methods

Lecture 14 Derivatives of linear functions, Exercise 1

Lecture 15 Derivatives of quadratic functions, Exercise 2

Lecture 16 Derivatives of quadratic functions, Exercise 3

Lecture 17 Derivative of a cubic polynomial, Exercise 4

Lecture 18 Derivative of the square root function, Exercise 5

Lecture 19 Another (equivalent) way of defining derivatives, Exercise 6

Lecture 20 A function that is not differentiable at some point, Exercise 7

Lecture 21 Absolute values and cusps: a generalisation of Exercise 7

Lecture 22 Yet another way of defining differentiability at a point

Lecture 23 Each differentiable function is continuous, but is the converse true?

Lecture 24 Optional: Proof of the part C1 from the theorem in Video 21

Lecture 25 Optional: Proof of the part C2.1 from the theorem in Video 21

Lecture 26 Is the absolute value always a bad news for global differentiability? Problem 1

Lecture 27 Derivatives of piecewise functions, Problem 2

Lecture 28 Recognising derivatives, Problem 3

Lecture 29 Recognising derivatives, Problem 4

Lecture 30 Recognising derivatives, Problem 5

Lecture 31 Computing derivatives from the definition, Problem 6

Lecture 32 One of my favourite problems, Problem 7

Lecture 33 Higher order derivatives, definition and notation

Lecture 34 Geometric interpretation of differentials

Lecture 35 What is linearization and why it is good for you

Lecture 36 Linearization works locally, Problem 8

Section 3: Deriving the derivatives of elementary functions

Lecture 37 Our plan

Lecture 38 The derivative of monic monomials (power functions 1), method 1

Lecture 39 The derivative of monic monomials (power functions 1), method 2

Lecture 40 The derivative of roots (power functions 2), method 1

Lecture 41 The derivative of power functions 3, method 1

Lecture 42 The derivative of sine, method 1

Lecture 43 The derivative of cosine, method 1

Lecture 44 The derivative of sine and cosine, method 2

Lecture 45 The derivative of sine inverse, method 1

Lecture 46 The derivative of cosine inverse

Lecture 47 The derivative of the exponential function

Lecture 48 The derivative of the natural logarithm, method 1

Lecture 49 The derivative of logarithms with any base

Lecture 50 Rules of differentiation: the main theorem

Lecture 51 An illustration for the Sum Rule

Lecture 52 Some illustrations for the Product Rule

Lecture 53 Three ways of writing a proof of the Sum Rule

Lecture 54 Three ways of writing a proof of the Scaling Rule

Lecture 55 Linearity of the differential operator, and its consequences

Lecture 56 A proof of the Product Rule

Lecture 57 Derivatives of polynomials are polynomials

Lecture 58 A proof of the Quotient (and Reciprocal) Rule

Lecture 59 The derivative of power functions 3, method 2

Lecture 60 Derivatives of rational functions are rational functions

Lecture 61 A generalization of the Product Rule

Lecture 62 The derivative of tangent

Lecture 63 The derivative of arctangent, method 1

Lecture 64 Some practice in differentiation, Exercise 1

Lecture 65 Some practice in differentiation, Exercise 2

Lecture 66 Some practice in differentiation, Exercise 3

Lecture 67 Some practice in differentiation, Exercise 4

Lecture 68 Some practice in differentiation, Exercise 5

Lecture 69 Some practice in differentiation, Exercise 6

Lecture 70 Some practice in differentiation, Exercise 7

Lecture 71 Some practice in differentiation, Exercise 8

Lecture 72 Some practice in differentiation, Exercise 9

Lecture 73 Some practice in differentiation, Exercise 10

Lecture 74 Differentiability of piecewise functions, Exercise 11

Lecture 75 The derivative of the sine of a scaled argument

Lecture 76 Differentiability of piecewise functions, Exercise 12

Lecture 77 Multiple zeros of polynomials and the round shapes of the graphs

Lecture 78 A really cool problem about polynomials, Problem 1

Lecture 79 The one with a picture, Problem 2

Lecture 80 Finding the tangent line, Problem 3

Lecture 81 Where to find more exercises for practice; some hints and tricks

Section 4: The Chain Rule and related rates

Lecture 82 About this section; some reading recommendations

Lecture 83 Repetition from Precalculus 1: compositions of functions

Lecture 84 Transformations of graphs that involve scalings of the argument

Lecture 85 The Chain Rule: the theorem, an example, and a proof

Lecture 86 A generalization of The Chain Rule and some related topics

Lecture 87 Back to Video 87 in Precalculus 1

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

Lecture 89 The derivative of power functions 3, method 3

Lecture 90 The derivative of power functions 4

Lecture 91 Some useful formulas from Precalculus 4

Lecture 92 The derivative of power functions 5

Lecture 93 The derivative of exponential functions

Lecture 94 Neither exponential nor power functions

Lecture 95 Back to some details from Videos 48 and 77

Lecture 96 The Chain Rule, an example

Lecture 97 How to handle differentiation in easy and complicated cases

Lecture 98 Some practice in differentiation (ChR), Exercise 1

Lecture 99 Some practice in differentiation (ChR), Exercise 2

Lecture 100 Some practice in differentiation (ChR), Exercise 3

Lecture 101 Some practice in differentiation (ChR), Exercise 4

Lecture 102 Some practice in differentiation (ChR), Exercise 5

Lecture 103 Some practice in differentiation (ChR), Exercise 6

Lecture 104 Some practice in differentiation (ChR), Exercise 7

Lecture 105 Some practice in differentiation (ChR), Exercise 8

Lecture 106 Some practice in differentiation (ChR), Exercise 9

Lecture 107 Some practice in differentiation (ChR), Exercise 10

Lecture 108 Derivatives of hyperbolic functions, Exercise 11

Lecture 109 Derivatives of inverse hyperbolic functions, Exercise 12

Lecture 110 Related rates and The Chain Rule

Lecture 111 Related Rates, Problem 1

Lecture 112 Related Rates, Problem 2

Lecture 113 Related Rates, Problem 3

Lecture 114 Related Rates, Problem 4

Lecture 115 Related Rates, Problem 5

Lecture 116 More practice: in the article and in the book

Section 5: Derivatives of inverse functions

Lecture 117 Derivative of functions inverse to differentiable functions, an intuition

Lecture 118 Derivative of functions inverse to differentiable functions, the theorem

Lecture 119 Derivative of functions inverse to differentiable functions, Example 1

Lecture 120 The derivative of an inverse, Example 2

Lecture 121 The derivative of roots (power functions 2), method 2

Lecture 122 The derivative of the natural logarithm, method 2

Lecture 123 The derivative of sine inverse, method 2

Lecture 124 The derivative of arctangent, method 2

Lecture 125 Optional: Intercept equations of straight line

Lecture 126 Optional: A theoretical statement about tangent lines

Lecture 127 The derivative of an inverse, Problem 1

Lecture 128 The derivative of an inverse, Problem 2

Lecture 129 The derivative of an inverse, Problem 3

Section 6: Mean value theorems and other important theorems

Lecture 130 Lots of theory that can be illustrated in a very intuitive way

Lecture 131 The concepts of maximum and minimum are not reserved for continuous functions

Lecture 132 The concept of monotonicity is not reserved for continuous functions

Lecture 133 Lemma about positive and negative derivatives

Lecture 134 Absolute values and cusps: proof of part C2.2 from V21

Lecture 135 Fermat's Theorem: Necessary condition for extremums at interior points

Lecture 136 Critical (stationary) points and singular points; plateaus

Lecture 137 Rolle's Theorem: About existence of a stationary point

Lecture 138 Rolle's Theorem, Example 1

Lecture 139 Rolle's Theorem, Example 2

Lecture 140 The Mean Value Theorem (Lagrange)

Lecture 141 Lagrange's Theorem, Example 1

Lecture 142 Lagrange's Theorem, Example 2

Lecture 143 Lagrange's Theorem, Example 3

Lecture 144 Extended Mean Value Theorem (Cauchy)

Lecture 145 Darboux property for derivatives

Lecture 146 Darboux property for derivatives, some examples

Lecture 147 About monotonicity of differentiable functions

Lecture 148 About monotonicity of differentiable functions, some examples

Lecture 149 Monotonicity of exponential functions

Lecture 150 Back to the derivatives of inverse functions, Problem 4

Lecture 151 Functions with derivative equal to zero

Lecture 152 Corollary about functions that have the same derivative on an interval

Lecture 153 Back to Video 106: explaining some subtleties in the Corollary in V152

Lecture 154 Back to Video 107: explaining some subtleties in the Corollary in V152

Lecture 155 Optional: Back to Video 107; a trigonometry-based solution

Section 7: Applications: monotonicity and optimisation

Lecture 156 You know everything you need to know about monotonicity and optimisation

Lecture 157 Polynomials: some examples from Precalculus 2 revisited

Lecture 158 How to find the vertex of a parabola if you hate completing the square

Lecture 159 A rational function, Problem 1 from V220 in Calc1p1

Lecture 160 The one with two square roots, Problem 2 from V221 in Calc1p1

Lecture 161 The one with arctangent, Problem 3 from V222 in Calc1p1

Lecture 162 Where the derivative doesn't help much, Problem 4 from V223 in Calc1p1

Lecture 163 The one with a one-sided vertical asymptote, Problem 5 from V224 in Calc1p1

Lecture 164 The First Derivative Test

Lecture 165 The Second Derivative Test

Lecture 166 Comparison between two tests: advantages and disadvantages

Lecture 167 Optimisation of continuous functions on compact and non-compact domains

Lecture 168 Optimisation, Problem 1

Lecture 169 Optimisation, Problem 2

Lecture 170 Optimisation, Problem 3

Lecture 171 Optimisation, Problem 4

Lecture 172 Optimisation, Problem 5

Lecture 173 Comparing numbers, Problem 6

Lecture 174 Comparing numbers, Problem 7

Section 8: Convexity and second derivatives

Lecture 175 Convexity, concavity, inflection points

Lecture 176 Convexity, concavity, inflection points: many examples

Lecture 177 Convexity, Problem 1

Lecture 178 Convexity, Problem 2

Lecture 179 Convexity, Problem 3

Section 9: l'Hôpital's rule with applications

Lecture 180 l'Hôpital's rule, the theorem with a proof (in an article)

Lecture 181 l'Hôpital's rule, Exercise 1

Lecture 182 l'Hôpital's rule, Exercise 2

Lecture 183 l'Hôpital's rule, Exercise 3

Lecture 184 l'Hôpital's rule, Exercise 4

Lecture 185 l'Hôpital's rule, Exercise 5

Lecture 186 l'Hôpital's rule, Exercise 6

Lecture 187 l'Hôpital's rule, Exercise 7

Lecture 188 l'Hôpital's rule, Exercise 8

Lecture 189 l'Hôpital's rule, Exercise 9

Section 10: Higher order derivatives and an intro to Taylor's formula

Lecture 190 Why we want to approximate functions with polynomials

Lecture 191 Monic monomials closely to zero

Lecture 192 Remember higher-order derivatives? Smooth and less smooth functions

Lecture 193 How to construct a polynomial that has the same derivatives at some point as f

Lecture 194 Maclaurin polynomial for the exponential function

Lecture 195 Maclaurin polynomial for the sine

Lecture 196 Maclaurin polynomial for cosine

Lecture 197 Taylor polynomial and the Second (and more!) Derivative Test

Lecture 198 Approximation, an example

Lecture 199 Limit, an example

Lecture 200 Order relation on the classes of functions introduced in V192

Section 11: Implicit differentiation

Lecture 201 Explicit versus implicit, Example 0

Lecture 202 Implicit differentiation and how it works

Lecture 203 One more example easy to handle both ways, Example 1

Lecture 204 Derivative of the logarithm, method 3

Lecture 205 The derivative of roots (power functions 2), method 3

Lecture 206 Derivative of arctangent, method 3

Lecture 207 Derivative of arcsine, method 3

Lecture 208 The one with a heart

Lecture 209 Plenty of problems to solve; we walk through two of them

Section 12: Logarithmic differentiation

Lecture 210 Logarithmic differentiation, Problem 1

Lecture 211 Logarithmic differentiation, Problem 2

Lecture 212 Logarithmic differentiation, Problem 3

Lecture 213 Logarithmic differentiation, Problem 4

Lecture 214 Logarithmic differentiation, Problem 5

Section 13: (Optional/advanced/future): Very briefly about partial derivatives

Lecture 215 Functions of several variables

Lecture 216 Partial derivatives, Exercise 1

Lecture 217 Higher-order partial derivatives, Exercise 2

Section 14: (Optional/advanced/future): Very briefly about antiderivatives

Lecture 218 Reverting differentiation

Lecture 219 A word about main integration techniques

Lecture 220 It is more useful than you think

Section 15: (Optional/advanced/future): A very brief introduction to the topic of ODE

Lecture 221 Various types of equations

Lecture 222 Differential equations mentioned in Precalculus 4

Lecture 223 Solving versus verifying solutions to ODE

Section 16: (Optional/future): More advanced concepts built upon the concept of derivative

Lecture 224 Plenty of derivative-like creatures

Lecture 225 Multivariate Taylor polynomials

Lecture 226 Mean Value Theorem used for functions of several variables

Section 17: Problem solving: optimisation

Lecture 227 Optimisation: a practical section

Lecture 228 Optimisation, Problem 1

Lecture 229 Optimisation, Problem 2

Lecture 230 Optimisation, Problem 3

Lecture 231 Optimisation, Problem 4

Lecture 232 Optimisation, Problem 5

Lecture 233 Optimisation, Problem 6

Lecture 234 Proving inequalities, Problem 7

Lecture 235 Proving inequalities, Problem 8

Section 18: Problem solving: plotting functions

Lecture 236 How to make a table of (sign) variations

Lecture 237 A brief repetition about asymptotes

Lecture 238 Plotting functions, Problem 1

Lecture 239 Plotting functions, Problem 2

Lecture 240 Plotting functions, Problem 3

Lecture 241 Plotting functions, Problem 4

Lecture 242 Plotting functions, Problem 5

Lecture 243 Plotting functions, Problem 6

Lecture 244 Plotting functions, Problem 7

Lecture 245 Wrap-up Calculus 1

Section 19: Extras

Lecture 246 Bonus Lecture

University and college students wanting to learn Single Variable Calculus (or Real Analysis),High school students curious about university mathematics; the course is intended for purchase by adults for these students