Derivative formulas through geometry | Chapter 3, Essence of calculus
Derivative formulas through geometry | Chapter 3, Essence of calculus

Making Calculus fun tolerable

Chris is a Stanford-educated tutor with over 10 years experience tutoring Calculus to students of all abilities, from students struggling to get from a C to a B, to go-getters trying to move an A- up to an A, to struggling students just hoping to pass. In that time he got a lot of experience learning how to explain this stuff in a way it actually makes sense to non-math people. Through his videos he has helped countless students, and he can do the same for you.

Calculus

Whether you’re in high school or college, AP or regular, AB or BC, “Calculus for Business” or “Calculus for Science & Engineering,” calculus classes always cover basically the same topics, in the same order. The only difference comes in whether certain topics are skipped and how hard the problems are. So no matter what school you’re in, if you’re in calculus, this page is for you! We sometimes get the question of where Calc 1 ends and Calc 2 begins… It’s different by school, but since everyone goes in roughly the same order, just look for where your class is now.
  1. I. Limits & Continuity:
  2. The videos in this chapter cover the more conceptual side of limits. In the first video we cover what limits are, and give an overview of the various types of limit problems you’ll see in calculus. The rest of the videos cover analyzing graphs for limits, figuring out if the limit “exists”, and finding the limits of piecewise functions.
  3. Epsilon-Delta Limit Problems

    What are Delta and Epsilon?
    Delta-Epsilon Problems as Slope Problems
    Delta-Epsilon: Linear Examples
    Delta-Epsilon: Non-Linear Examples (Different from left and right)
    Examples Given Epsilon
    Delta-Epsilon Problems: Crazy Wording

    If your teacher covers this, you’ll know it! Very painful way to start off limits.
  4. This chapter covers how to figure out if a function is continuous from both a graphical and limit perspective, including the definition of continuous.
  5. One-sided limits are the ones where you’re approaching x from either the right or the left. 3+ means limit approaching 3 from the right, 3- means approaching 3 from the left.
  6. Limits of Giant Fractions (Rational Functions)

    Limits of Polynomial Fractions: Plugging in
    Limits of Polynomial Fractions: Indeterminate, Infinite, and “Does Not Exist” solutions
    Limits of Polynomial Fractions: Difference of Cubes
    Limits of Polynomial Fractions: U-Substitution
    Limits With Roots & Radicals

    These are your classic “big mess of algebra” limit problems, which happen when a limit is “indeteriminate” (plugging in results in 0/0 or infinity/infinity). This chapter covers finding limits of “giant fractions” (i.e. rational expressions) containing polynomials and roots & radicals.
  7. This chapter covers what to do when X is approaching infinity (the sideways 8 symbol), as well as how we can use this new skill to find horizontal asymptotes of rational functions.
  8. You’ll recognize these when you see them. They’ll give you the limit of f & g (but not the equations themselves) and make you combine them. Sort of like log rules, if you’re into that.
  9. If a limit has a sine or cosine, this chapter covers it. Besides knowing your unit circle so you can plug in, this chapter has a couple special formulas and strategies. Also covered are special limits like sinx/x and cosx/x.
  10. These nasty puppies are limit problems with a sine or cosine which has an X in the denominator of the “argument”, like sin(1/x) or cos(pi/x).
  11. A theorem that’s in the top five of most useless things you’ll learn (or not) in calculus. Very little use, unless your teacher tells you it’s on the test.
  12. Limit Definition of Derivative

    (a.k.a. Difference Quotient)
    Limit Definition of Derivatives (a.k.a. Difference Quotient)
    Hard Limit Definition of Derivatives Problems

    These videos introduce the limit definition of derivatives, which every class covers and then forgets about. It’s the one where you have to find f(x+h), then somehow plug in h and take the limit as h approaches zero.
  13. This shortcut for finding limits is easier than everything that’s come before, but it requires taking derivatives.
  14. II. Derivatives
  15. A tangent line is the equation of a line that’s tangent to a function at a particular point, and you find it by using derivatives. This line is important because it’s slope is the “rate of change” of the function at that point. Just to make things awesome, we’ll also review point-slope form of lines since that’s the easiest way to find a tangent line.
  16. This chapter covers the formulas for taking the derivatives of exponents, polynomials, powers of x, trig functions (sin, cos, tan, cot, sec, csc), exponentials, and radicals. As long as you don’t need chain rule.
  17. Product & Quotient Rules (no chain rule)

    Product Rule (without chain rule)
    Quotient Rule (without chain rule)

    The basics of the product and quotient rules, but without the chain rule. There’s a similar chapter covering these formulas with the chain rule later.
  18. Using The Chain Rule to take derivatives of:

    Power Rule
    Roots & Radicals
    Trig Functions
    Inverse Trig Functions
    Exponentials
    Logs

    In this lengthy chapter we’ll re-learn all the derivative formulas, except this time using the Chain Rule too: exponents (power rule), roots & radicals, trig functions, inverse trig functions, exponentials, and natural logs. A must-watch for Calculus students!
  19. Product & Quotient Rules (WITH chain rule)

    Product Rule With Chain Rule
    Quotient Rule With Chain Rule

    This chapter brings the chain rule to product and quotient derivatives. If you haven’t, check out the first product & quotient rule chapter first.
  20. Implicit Differentiation

    What The Heck Is Implicit Differentiation?
    Finding Tangent Lines Using Implicit Differentiation
    Finding Second Derivatives With Implicit Differentiation

    In this chapter we’ll cover the basics of taking derivatives implicitly (finding y’), using them to find equations of tangent lines, and finding second derivatives (y”).
  21. Related Rates

    What The Heck Are Related Rates?
    Basic Related Rates Word Problems
    Triangle-Based Word Problems
    Area Related Rates Word Problems (AP level)
    Volume Related Rates Word Problems (AP level)

    Derivative word problems involving rates of change in area, volume, etc. All derivatives taken with respect to time.
  22. Mean Value & Rolle’s Theorem

    Instantaneous Rate of Change
    Average Rate of Change

    In this chapter we cover these two straightforward (but basically useless) problem types that every teacher seems to ask.
  23. Graphing Derivatives

    Derivatives Graphing Overview
    Sketching Derivatives Of Functions
    How To Find Critical Values
    Intervals Of Increase & Decrease

    This chapter is a grab bag of graphical analysis. Intervals of increase and decrease, how to find critical values, how to sketch the derivative of a function just from the sketch of the original function, and a general intro to relative extrema (maxima and minima).
  24. Maxima, Minima, &

    “The First Derivative Test”
    Relative Maxima & Minima: First derivative Test
    Relative versus Absolute
    Relative Maxima & Minima:Easy and Hard Examples
    Absolute Extrema

    Extrema (maximums and minimums) come in two flavors: relative and absolute. This chapter covers both, and how to find them using the first derivative test.
  25. Optimization Word Problems

    Projectile Word Problems
    Maximizing Area Word Problems
    Maximizing Volume Word Problems
    Optimization Word Problems: Fool Proof Step by Step

    The “other” derivative word problems (related rates are the big one), where you maximize or minimize the area, volume, or cost of some quantity.
  26. Inflection Points & Curve Sketching

    Inflection Points, Concavity & Second Derivative Test
    Curve Sketching Examples
    Sketching Parent Graphs from Derivative Graphs
    Review: Vertical Asymptotes & Holes
    Review: Using Limits To Find Horizontal Asymptotes
    Review: Slant Asymptotes (a.k.a. Oblique Asymptotes)
    Review: Relative Maxima & Minima
    Review: Finding X and Y Intercepts

    This chapter introduces concavity, points of inflection and the second derivative test, then reviews asymptotes, relative extrema, and how to find intercepts so you’ll have the tools for graphing functions calculus-style.
  27. Unique Derivatives

    Logarithmic Differntiation aka Derivatives with Variables In Both Exponent & Base

    The videos in this chapter cover specific and unusual types of derivatives that I wasn’t sure where else to put, such as: xx and (sinX)x.
  28. Linear Approximation & Differentials

    What’s A Linear Approximation
    Linear Approximation Examples
    Differentials

    Using the equation of the tangent line to approximate values of functions.
  29. Euler’s Method & Slope Fields

    Euler’s Method: What, Why, and How
    Euler’s Method: Word Problems
    Euler’s Method vs Slope Fields

    Slope fields and Euler’s Method are actually pretty similar, so the videos in this chapter explain how to manage both.
  30. Newton’s Method

    Newton’s Method of Approximating Zeros
    Newton’s Method: Approximating Intersections of Functions
    Newton’s Method: No X-intercepts (ambiguous cases)
    Newton’s Method: Bad First Estimates

    A plug-and-chug chart for approximating the zeros (a.k.a. roots, intercepts, solutions) of a function.
  31. Differential Equations

    Verifying Solutions of Differential Equations & Solving For C
    Solving Differential Equations Using Separation of Variables

    This chapter covers solving differential equations using integration: separation of variables, initial conditions, general solutions, specific solutions.
  32. The video in this chapter explains the basics of partial derivatives, how to find them, their notation, and how to tell if a function is continuous based on its partial derivatives.
  33. III. Integrals & Anti-Derivatives
  34. Intro to Integration & Anti-Derivatives

    The Power Rule (no U-Substitution)
    Why The Heck Do We Need +C?
    Integrating Roots Radicals & Fractions (no U-Sub)
    Trig Function Integrals (without U-Substitution)
    Log & Exponential Integrals (without U-Substitution)

    Basic integration (antiderivatives) of power rule polynomials, roots & radicals, trig functions, and initial value problems, all WITHOUT u-substitution.
  35. Riemann Sums & Area

    Left Sum vs Right Sum vs Upper Bound vs Lower Bound
    A Few More Area Approximation Examples
    The Trapezoid Rule For Approximating Area Under A Curve

    In this chapter we’ll approximate area using left-hand sums, right-hand sums, midpoint, upper bounds, lower bounds, and trapezoidal rules. Collectively, these are called “Riemann Sums” or “approximation integration”.
  36. Definite Integrals

    Definite Integrals & The Fundamental Theorem of Calculus
    Positive vs Negative Area
    Integral Properties

    In this chapter we use The Fundamental Theorem of Calculus and definite integrals (the ones with little numbers on the integral sign) to find the area under curves, negative area, and integral properties.
  37. U-Substitution Integration:

    Power Rule & Exponents
    Roots & Radicals
    Trig Functions
    Exponentials
    Logs

    This chapter covers U-substitution with all the major integral types: power rule, roots, radicals, rational functions, fractions, exponentials, logs, trig functions.
  38. Average Value & Mean Value Theorem

    Average Value and The Mean Value Theorem
    Average Value Word Problems

    Finding the average value of a function on an interval, word problems, plug-and-chug.
  39. Fundamental Theorem of Calculus

    Definite Integrals & The Fundamental Theorem of Calculus
    Second Fundamental Theorem of Calculus

    Both first and second fundamental theorems.
  40. Using dx and dy integrals to find the area between functions
  41. Disks, Washers & Shells

    What the heck are revolved solids?!
    Overview: Disks vs Washers vs Shells
    The Disk Method
    Washer Method
    Shell Method (they should be called “tubes”)
    Disks vs Shells: picking your poison
    Volume of Swept Solids

    Volume: the hardest topic in calculus. Four hours of videos get you through the integral disk method, washer method, and shell method.
  42. Linear Approximation & Differentials

    What’s A Linear Approximation
    Linear Approximation Examples
    Differentials

    Using the equation of the tangent line to approximate values of functions.
  43. IV. Advanced Integration Techniques
  44. Integration By Parts

    Intro to Integration By Parts
    Integration By Parts: Natural Logs
    Advanced Integration By Parts Problems

    From easy to hard, this formula allows us to use u, v, du and dv to integrate the products of functions multiplied together.
  45. Integration by Trigonometric Substitution

    Trig Substitution (Step 1): choosing what to sub
    Trig Substitution (Step 2): how to un-substitute theta for x
    Trig Substitution Examples

    Also called “trig sub”, a method for solving integrals with square roots in them by substituting a trig function for x.
  46. Trig Function Integrals

    Power Reducing Formulas for Sine & Cosine
    Integrating Powers of Sine and Cosine
    Integrating Powers of Tan & Cot
    Integrating Powers of Secant & Co-Secant (with Tan & Cot)

    The power reducing formula for sine and cosine, as well as how to integrate powers of sine, cosine, tangent, co-tangent, secant, co-secant.
  47. Partial Fraction Decomposition

    Integration by Partial Fraction Decomposition
    Partial Fractions with “Non-Repeated Linear Factors”
    Partial Fractions: Repeat & Non-Linear Factors

    This chapter reviews partial fraction decomposition (in case you’re rusty), then goes through how to use the technique to integrate some nasty rational functions.
  48. Improper Integrals

    Improper Integrals: Upper or Lower Bound Is Infinity
    Improper Integrals with Discontinuities

    Integrals where infinity is one of your limits of integration, or the function doesn’t exist at one of the limits.
  49. Using integration to find arc length of a function, or surface area of a revolved surface, on an interval.
  50. Differential Equations

    Verifying Solutions of Differential Equations & Solving For C
    Solving Differential Equations Using Separation of Variables

    This chapter covers solving differential equations using integration: separation of variables, initial conditions, general solutions, specific solutions.
  51. Parametric Equations

    Graphing Parametric Equations
    Converting Parametric Equations to Rectangular
    Converting Rectangular Equations to Parametric
    Parametric Equations of Conic Sections: Circles, Ellipses & Hyperbolas

    This chapter covers converting parametric equations to rectangular and back again, eliminating the parameter, parametric forms of circles and ellipses, and graphing them.
  52. V. Business Calculus
  53. These videos cover a few topics that are taught in Business Calculus and Economics, which also show up sometimes in regular calculus classes. Marginal cost, marginal revenue, and marginal profit. The Demand Function, which gives you the demand (x) based on price (p). Overall, these topics allow you to calculate and maximize the profit of a business.
  54. VI. Physics Applications of Calculus
  55. Work Done By A Force

    Work Done By A Constant Force
    What the heck is work?
    Work Done By A Variable Force (integrals)

    Work by a constant or variable force: gravity, expanding gas, friction, efficiency, power. Also fun explanations of what the heck work is, and how to figure out what these problems are even asking!
  56. Centroids & Center of Mass

    One-Dimensional Centroids
    Two-Dimensional Moments & Centroids
    Centroids Using Integration

    The first two videos in this chapter cover finding center of mass of one-dimensional and two-dimensional (2-D) systems without using calculus, then the final video covers using integrals.
  57. Fluid Pressure & Forces

    Intro to Pressure: Concepts, Formulas & Units
    Pressure & Force Due to Depth
    Using Integrals to Calculate Pressure Force

    This chapter covers the basics of pressure and the forces that pressure can exert on a surface, then it gets into using integrals to find the pressure on a vertical surface (plate, window, etc.).
  58. Vectors

    Dealing With Vectors Graphically
    Adding, Subtracting & Multiplying Vectors
    The Dot Product and Scalar Product
    Magnitude & Unit Vectors
    Components of Vectors
    i, j, k Notation
    Cross Product of Vectors

    Basics of vector addition, subtraction, multiplication, dot product, scalar product, magnitude, unit vectors, cross multiplication, and components.
  59. Projectile Motion

    (Pre-Calc Version)
    One-Dimensional Gravity Problems
    Two-Dimensional Projectile Problems

    This chapter covers kinematics projectile motion problems as you would see in Pre-Calculus or Algebra 2 math classes. This topic is covered in more depth on the physics page. One-dimensional and two-dimensional gravity problems, range, vector components of velocity, etc.
  60. VII. Series
  61. Sequences, Series & Sigma Notation

    Intro to Sequences and Series
    Arithmetic Sequences & Series
    Geometric Sequences & Series
    Sigma Notation
    SAT-type Sequence Questions (Word Problems)

    Common confusion: a “series” is just a sequence with plus signs between the terms instead of commas. All other questions, check out the chapter page, which includes a free printable pdf of all the formulas for arithmetic and geometric sequences.
  62. Convergence & Divergence:

    Nth Term Test
    Geometric Series Test
    P-Series Test
    Integral Test
    Comparison Test
    Limit Test
    Ratio Test
    Alternating Series

    Everything you could possibly need to know about convergence and divergence of infinite series is all in this one chapter because otherwise it would be really confusing. In addition to the topics listed to the left, there’s a free overview of series convergence, as well as more obscure topics like the remainder (error) of an alternating series approximation.
  63. Taylor & Maclaurin Series

    Intro to Taylor & Maclaurin Series
    Taylor & Maclaurin Series Examples
    Approximating Functions With Taylor & Maclaurin Series

    This chapter covers non-linear approximation of functions, and the series expansions which make them possible.
  64. This chapter covers radius of convergence and intervals of convergence for power series, which are just the generic name for infinite series like Taylor and Maclaurin.

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