Differential Calculus
Learn differential calculus—limits, continuity, derivatives, and derivative applications.
6 units252 videos26 hours
Learn it live with a class on Puzzicle · Teachers: use it with your class
Unit 1: Limits and continuity
Limits intro
Estimating limits from graphs
Estimating limit values from graphs ✓ Check-ins6:01
Examining unbounded limits ✓ Check-ins2:32
Estimating one-sided limits from graphs ✓ Check-ins9:11
Estimating one-sided limits from graphs: asymptote ✓ Check-ins2:27
Connecting limits and graphical behavior ✓ Check-ins5:03
Connecting limits and graphical behavior (more examples) ✓ Check-ins5:16
Estimating limits from tables
Formal definition of limits (epsilon-delta)
Exploring the formal definition of limits Part 1: intuition review ✓ Check-ins3:36
Exploring the formal definition of limits Part 2: building the idea ✓ Check-ins6:06
Exploring the formal definition of limits Part 3: the definition ✓ Check-ins ✎ Fixed a slip6:59
Exploring the formal definition of limits Part 4: using the definition ✓ Check-ins8:30
Properties of limits
Introducing properties of limits ✓ Check-ins5:08
Determining limits of combined functions ✓ Check-ins4:09
Determining limits of combined functions: piecewise functions ✓ Check-ins4:13
Examining the theorem for limits of composite functions ✓ Check-ins3:13
Examining the theorem for limits of composite functions: when conditions aren't met ✓ Check-ins3:22
Determining limits of composite functions: internal limit doesn't exist ✓ Check-ins4:17
Determining limits of composite functions: external limit doesn't exist ✓ Check-ins3:25
Limits by direct substitution
Finding limits by direct substitution ✓ Check-ins2:07
Examining undefined limits by direct substitution ✓ Check-ins2:14
Determining limits of trigonometric functions ✓ Check-ins6:06
Determining limits of piecewise functions ✓ Check-ins3:49
Determining limits of piecewise functions: absolute value ✓ Check-ins6:20
Limits using algebraic manipulation
Strategy in finding limits
Squeeze theorem
Types of discontinuities
Continuity at a point
Continuity over an interval
Removing discontinuities
Infinite limits
Limits at infinity
Introducing limits at infinity ✓ Check-ins2:46
Exploring functions with same limit at infinity ✓ Check-ins3:46
Determining limits at infinity of quotients (Part 1) ✓ Check-ins4:07
Determining limits at infinity of quotients (Part 2) ✓ Check-ins4:33
Determining limits at infinity of quotients with square roots (odd power) ✓ Check-ins5:14
Determining limits at infinity of quotients with square roots (even power) ✓ Check-ins3:37
Limits at infinity of quotients with trig ✓ Check-ins3:04
Limits at infinity of quotients with trig (limit undefined) ✓ Check-ins2:21
Limit at infinity of a difference of functions ✓ Check-ins5:26
Unit 2: Derivatives: definition and basic rules
Average vs. instantaneous rate of change
Connecting Newton, Leibniz, and Usain Bolt ✓ Check-ins9:07
Exploring derivatives as a concept ✓ Check-ins7:16
Connecting secant lines and average rate of change ✓ Check-ins5:17
Exploring the derivative as slope of a curve ✓ Check-ins6:10
Connecting the derivative and tangent line equations ✓ Check-ins7:33
Secant lines
Slope of a line secant to a curve ✓ Check-ins7:17
Secant line with arbitrary difference ✓ Check-ins4:19
Secant line with arbitrary point ✓ Check-ins ✎ Fixed a slip4:19
Secant line with arbitrary difference (with simplification) ✓ Check-ins3:54
Secant line with arbitrary point (with simplification) ✓ Check-ins3:52
Secant lines: challenging problem 1 ✓ Check-ins ✎ Fixed a slip6:18
Secant lines: challenging problem 2 ✓ Check-ins5:57
Derivative definition
Exploring the formal definition of the derivative as a limit ✓ Check-ins15:43
Understanding the formal and alternate form of the derivative ✓ Check-ins4:53
Working an example: Derivative as a limit ✓ Check-ins5:46
Working an example: Derivative from limit expression ✓ Check-ins5:17
Finding the derivative of x² at x=3 using the formal definition ✓ Check-ins8:28
Finding the derivative of x² at any point using the formal definition ✓ Check-ins ✎ Fixed a slip11:05
Limit expression for the derivative of function (graphical) ✓ Check-ins3:46
Estimating derivatives
Differentiability
Exploring differentiability and continuity ✓ Check-ins9:38
Understanding differentiability at a point: graphical ✓ Check-ins5:39
Examining differentiability at a point: algebraic (function is differentiable) ✓ Check-ins5:02
Examining differentiability at a point: algebraic (function isn't differentiable) ✓ Check-ins6:22
Power rule
Derivative rules: constant, sum, difference, and constant multiple
Combining the power rule with other derivative rules
Derivatives of cos(x), sin(x), 𝑒ˣ, and ln(x)
Product rule
Quotient rule
Derivatives of tan(x), cot(x), sec(x), and csc(x)
Proof videos
Exploring the proof: Differentiability implies continuity ✓ Check-ins11:40
Justifying the power rule ✓ Check-ins6:42
Proving the power rule for positive integer powers ✓ Check-ins ✎ Fixed a slip7:03
Proving the power rule for the square root function ✓ Check-ins5:07
Determining the limit of sin(x)/x as x approaches 0 ✓ Check-ins9:16
Determining the limit of (1-cos(x))/x as x approaches 0 ✓ Check-ins4:05
Proving the derivative of sin(x) ✓ Check-ins5:52
Proving the derivative of cos(x) ✓ Check-ins3:18
Proving the product rule ✓ Check-ins9:26
Unit 3: Derivatives: chain rule and other advanced topics
Chain rule
Using the chain rule ✓ Check-ins5:07
Examining common chain rule misunderstandings ✓ Check-ins6:48
Identifying composite functions ✓ Check-ins8:13
Working an example: Derivative of cos³(x) using the chain rule ✓ Check-ins9:17
Working an example: Derivative of √(3x²-x) using the chain rule ✓ Check-ins ✎ Fixed a slip5:31
Working an example: Derivative of ln(√x) using the chain rule ✓ Check-ins6:45
Working an example: Chain rule with table ✓ Check-ins2:27
More chain rule practice
Determining the derivative of aˣ (for any positive base a) ✓ Check-ins5:25
Determining the derivative of logₐx (for any positive base a≠1) ✓ Check-ins4:48
Working an example: Derivative of 7^(x²-x) using the chain rule ✓ Check-ins3:40
Working an example: Derivative of log₄(x²+x) using the chain rule ✓ Check-ins4:10
Working an example: Derivative of sec(3π/2-x) using the chain rule ✓ Check-ins7:11
Working an example: Derivative of ∜(x³+4x²+7) using the chain rule ✓ Check-ins ✎ Fixed a slip5:55
Implicit differentiation
Implicit differentiation (advanced examples)
Differentiating inverse functions
Derivatives of inverse trigonometric functions
Strategy in differentiating functions
Differentiation using multiple rules
Differentiating using multiple rules: strategy ✓ Check-ins5:47
Applying the chain rule and product rule ✓ Check-ins5:54
Applying the chain rule twice ✓ Check-ins3:10
Determining the derivative of eᶜᵒˢˣ⋅cos(eˣ) ✓ Check-ins5:52
Determining the derivative of sin(ln(x²)) ✓ Check-ins4:52
Product rule to find derivative of product of three functions ✓ Check-ins3:34
Second derivatives
Disguised derivatives
Logarithmic differentiation
Unit 4: Applications of derivatives
Meaning of the derivative in context
Straight-line motion
Introducing one-dimensional motion with calculus ✓ Check-ins6:31
Interpreting direction of motion from position-time graph ✓ Check-ins2:39
Interpreting direction of motion from velocity-time graph ✓ Check-ins1:46
Interpreting change in speed from velocity-time graph ✓ Check-ins3:57
Working an example: Motion problems with derivatives ✓ Check-ins4:58
Total distance traveled with derivatives ✓ Check-ins10:09
Non-motion applications of derivatives
Introduction to related rates
Introducing related rates ✓ Check-ins7:43
Analyzing related rates problems: expressions ✓ Check-ins8:07
Analyzing related rates problems: equations (Pythagoras) ✓ Check-ins3:35
Analyzing related rates problems: equations (trig) ✓ Check-ins8:01
Differentiating related functions intro ✓ Check-ins ✎ Fixed a slip3:24
Working an example: Differentiating related functions ✓ Check-ins7:52
Solving related rates problems
Approximation with local linearity
L’Hôpital’s rule
Introducing L'Hôpital's rule ✓ Check-ins8:51
Applying L'Hôpital's rule: limit at 0 example ✓ Check-ins ✎ Fixed a slip7:43
Applying L'Hôpital's rule: limit at infinity example ✓ Check-ins5:15
L'Hôpital's rule: challenging problem ✓ Check-ins7:50
L'Hôpital's rule: solve for a variable ✓ Check-ins4:10
Proving a special case of l'Hôpital's rule ✓ Check-ins5:21
L’Hôpital’s rule: composite exponential functions
Unit 5: Analyzing functions
Mean value theorem
Introducing the mean value theorem ✓ Check-ins6:37
Applying the mean value theorem example: polynomial ✓ Check-ins4:50
Applying the mean value theorem example: square root function ✓ Check-ins6:24
Justifying use of the mean value theorem: table ✓ Check-ins3:41
Justifying use of the mean value theorem: equation ✓ Check-ins2:54
Applying the mean value theorem ✓ Check-ins7:12
Extreme value theorem and critical points
Intervals on which a function is increasing or decreasing
Relative (local) extrema
Introducing minimum and maximum points ✓ Check-ins5:30
Finding relative extrema (first derivative test) ✓ Check-ins5:25
Working an example: finding relative extrema ✓ Check-ins7:49
Analyzing mistakes when finding extrema (example 1) ✓ Check-ins4:33
Analyzing mistakes when finding extrema (example 2) ✓ Check-ins4:42
Absolute (global) extrema
Concavity and inflection points intro
Analyzing concavity and inflection points
Analyzing concavity (algebraic) ✓ Check-ins ✎ Fixed a slip9:16
Analyzing inflection points (algebraic) ✓ Check-ins ✎ Fixed a slip5:35
Exploring mistakes when finding inflection points: second derivative undefined ✓ Check-ins6:11
Exploring mistakes when finding inflection points: not checking candidates ✓ Check-ins4:02
Second derivative test
Sketching curves
Connecting f, f', and f''
Establishing a calculus-based justification for a function increasing ✓ Check-ins3:12
Justifying a function's features using the first derivative ✓ Check-ins5:53
Identifying inflection points from graphs of function and derivatives ✓ Check-ins5:06
Justifying a function's features using the second derivative: inflection point ✓ Check-ins3:47
Justifying a function's features using the second derivative: maximum point ✓ Check-ins ✎ Fixed a slip4:14
Connecting f, f', and f'' graphically ✓ Check-ins5:36
Connecting f, f', and f'' graphically (another example) ✓ Check-ins4:21
Solving optimization problems
Applying optimization: sum of squares ✓ Check-ins ✎ Fixed a slip7:35
Applying optimization: box volume (Part 1) ✓ Check-ins9:50
Applying optimization: box volume (Part 2) ✓ Check-ins9:00
Applying optimization: profit ✓ Check-ins ✎ Fixed a slip11:27
Applying optimization: cost of materials ✓ Check-ins ✎ Fixed a slip12:40
Applying optimization: area of triangle & square (Part 1) ✓ Check-ins ✎ Fixed a slip6:19
Applying optimization: area of triangle & square (Part 2) ✓ Check-ins5:31
Optimization problem: extreme normaline to y=x² ✓ Check-ins ✎ Fixed a slip23:13
Analyzing motion problems: finding the maximum acceleration ✓ Check-ins4:25
Analyzing implicit relations
Unit 6: Parametric equations, polar coordinates, and vector-valued functions
Parametric equations intro
Second derivatives of parametric equations
Vector-valued functions
Planar motion
Learn this with a class
With a class on Puzzicle this course runs as a timetable of live study parties: your batch watches together and answers the check-ins, sometimes with a teacher leading.
Course content © Khan Academy, used under CC BY-NC-SA. See this course on Khan Academy.