Duration of the online course: 7 hours and 33 minutes
New
Build ML-ready calculus skills fast with this free online course—master limits, derivatives, gradients, and integrals with practical exercises and autodiff tools.
In this free course, learn about
Compute and interpret limits, incl. cases where one-sided limits diverge
Link limits to derivatives via the difference quotient; interpret tangent slope
Use derivative notation clearly (dy/dx, d/dx operator) and choose appropriate forms
Use derivatives/gradients to update ML parameters via gradient descent
Understand and apply reverse-mode automatic differentiation (PyTorch/TensorFlow)
Represent computations as tensor graphs; enable gradients for model parameters
Compute partial derivatives and use multivariate chain rule in multivariable functions
Compute gradients of quadratic cost and mean squared error for linear regression
Understand backpropagation as efficient gradient computation through networks
Compute higher-order and mixed partial derivatives for multivariate functions
Understand integrals as accumulation/area; method of exhaustion and infinitesimals
Apply integration rules to find indefinite/definite integrals; do numeric integration
Use integral calculus to compute AUC/ROC via trapezoidal rule; interpret binary metrics
Course Description
Machine learning models do not learn by magic; they learn by measuring error and systematically reducing it. That process is calculus in action. This course connects the essential ideas of single-variable and multivariable calculus directly to how modern AI systems are trained, so the symbols on the page become tools you can recognize inside gradient descent, backpropagation, and everyday model debugging.
You will develop intuition for limits and what they reveal about behavior near a point, then translate that intuition into derivatives as rates of change and slopes of tangent lines. From there, the course builds the core differentiation rules that power optimization, showing how to move from a function to its gradient efficiently and correctly. Instead of treating formulas as isolated tricks, you learn why they work, how to choose the right rule for a given expression, and how those choices map to the way learning algorithms adjust parameters to reduce loss.
As the course advances into partial derivatives and the multivariate chain rule, you will see how training objectives depend on many parameters at once, and why the direction and sign of a gradient determine the next update step. You will also bridge the gap between theory and practice through automatic differentiation, the engine beneath libraries such as PyTorch and TensorFlow. By understanding how computation graphs represent functions and how reverse-mode differentiation scales to large models, you gain a clearer mental model of what your framework is doing when it computes gradients.
Integral calculus appears not as an abstract ending, but as a practical way to quantify accumulation and area. The course ties integration to evaluation concepts such as measuring performance across thresholds, including intuition for area under curves and numerical methods that approximate integrals when data comes in discrete points.
With frequent exercises, this course helps you become more confident reading ML papers, implementing training loops, and reasoning about optimization behavior. If you have ever used an AI library without feeling grounded in the math beneath it, this is the bridge from calculus fundamentals to machine learning competence.
Course content
Video class: Calculus I: Limits03m
Exercise: Why are derivatives central to training in machine learning models?
Video class: Intro to Differential Calculus —Topic 42 of Machine Learning Foundations13m
Exercise: On a distance versus time curve for a vehicle, what does the slope of the tangent line at a point represent?
Video class: Intro to Integral Calculus –Topic 43 of Machine Learning Foundations02m
Exercise: Differential vs. Integral Calculus in Motion
Video class: The Method of Exhaustion – Topic 44 of Machine Learning Foundations06m
Exercise: How does the method of exhaustion illustrate the core idea behind integrals used in machine learning?
Video class: Calculus of the Infinitesimals – Topic 45 of Machine Learning Foundations10m
Exercise: When you zoom in on a curve at a point until it looks straight to estimate its slope, which calculus concept are you applying?
Video class: Calculus Applications – Topic 46 of Machine Learning Foundations08m
Exercise: How is differential calculus primarily used when training ML models?
Video class: Calculating Limits – Topic 47 of Machine Learning Foundations18m
Exercise: Evaluate the limit as x approaches 1: (x^2 - 1) / (x - 1)
Video class: Exercises on Limits – Topic 48 of Machine Learning Foundations01m
Exercise: Two sided limit when one sided limits diverge with opposite signs
Video class: Derivatives and Differentiation — Segment 2 of Subject 3, Limits01m
Exercise: Which expression is the differentiation equation linking limits to derivatives
Video class: The Delta Method – Topic 49 of Machine Learning Foundations15m
Exercise: Using the delta method, what is the slope of the tangent to f(x)=x^2+2x+2 at x=2?
Video class: How Derivatives Arise from Limits – Topic 50 of Machine Learning Foundations14m
Exercise: Using the limit definition of the derivative, what is f'(2) for f(x)=x^2+2x+2?
Video class: Derivative Notation — Topic 51 of Machine Learning Foundations04m
Exercise: Which derivative notation most clearly identifies both dependent and independent variables for the first derivative?
Video class: The Derivative of a Constant –Topic 52 of Machine Learning Foundations01m
Exercise: Derivative of a constant in ML calculus
Video class: The Power Rule for Derivatives – Topic 53 of Machine Learning Foundations01m
Exercise: Using the power rule, what is d/dx of x^7?
Video class: The Constant Multiple Rule for Derivatives — Topic 54 of Machine Learning Foundations03m
Exercise: Apply the constant multiple rule: If g(x) = x^4, what is d/dx [3 g(x)]?
Video class: The Sum Rule for Derivatives — Topic 55 of Machine Learning Foundations02m
Exercise: Using the sum rule and power rule, what is d/dx of x^4 + x^9
Video class: Exercises on Derivative Rules — Topic 56 of Machine Learning Foundations03m
Exercise: Find the derivative of f(x) = 10x^5 - 6x^3 - x using basic rules
Video class: The Product Rule for Derivatives — Topic 57 of Machine Learning Foundations03m
Exercise: Compute dy/dx using the product rule for y = (6x^3)(7x^4)
Video class: The Quotient Rule for Derivatives — Topic 58 of Machine Learning Foundations04m
Exercise: Select the correct quotient rule form for y equals w over z with w equals 4x^2 and z equals x^3 plus 1
Video class: The Chain Rule for Derivatives — Topic 59 of Machine Learning Foundations06m
Exercise: Using the chain rule, compute dy/dx for y = (2x^2 + 8)^2
Video class: Advanced Exercises on Derivative Rules — Topic 60 of Machine Learning Foundations02m
Exercise: Which rule should you apply first to differentiate f(x) = (3x^2 + 1) sqrt(5x - 2)?
Video class: The Power Rule on a Function Chain — Topic 61 of Machine Learning Foundations05m
Exercise: Find dy dx for y equals left parenthesis 3x plus 1 right parenthesis squared using the power rule on a function chain
Video class: Automatic Differentiation – Segment 3 of Subject 3, Limits01m
Exercise: Which technique enables scalable derivative computation for large machine learning models and is implemented in libraries such as PyTorch and TensorFlow?
Video class: What Automatic Differentiation Is — Topic 62 of Machine Learning Foundations04m
Exercise: What best characterizes reverse mode automatic differentiation in calculus for machine learning
Video class: Automatic Differentiation with PyTorch — Topic 63 of Machine Learning Foundations06m
Exercise: What is dy/dx at x = 5 for y = x^2 computed via automatic differentiation?
Video class: Automatic Differentiation with TensorFlow — Topic 64 of Machine Learning Foundations03m
Exercise: Autodiff in TensorFlow: dy dx for y equals x squared at x equals 5
Video class: The Line Equation as a Tensor Graph — Topic 65 of Machine Learning Foundations20m
Exercise: Which action enables gradient computation for parameters in a PyTorch linear model y=mx+b
Video class: Machine Learning from First Principles, with PyTorch AutoDiff — Topic 66 of ML Foundations40m
Exercise: In a four step learning loop for linear regression with y_hat equals m x plus b, what is the role of step three?
Video class: Calculus II: Partial Derivatives22m
Exercise: In linear regression y = m x + b trained with mean squared error, if ∂C/∂m > 0 and ∂C/∂b > 0 at the current parameters, which update will reduce the cost using gradient descent?
Video class: What Partial Derivatives Are (Hands-on Introduction) — Topic 67 of Machine Learning Foundations29m
Exercise: Compute the slope of z with respect to y at y = -1 for z = x^2 - y^2 (x held constant)
Video class: Partial Derivative Exercises — Topic 68 of Machine Learning Foundations03m
Exercise: For z = x^2 − y^2, at x = 3 and y = 0, what is the partial derivative ∂z/∂x?
Video class: Calculating Partial Derivatives with PyTorch AutoDiff — Topic 69 of Machine Learning Foundations05m
Exercise: Using automatic differentiation for z = x^2 − y^2, what are the partial derivatives at (x, y) = (0, 0)?
Video class: Advanced Partial Derivatives — Topic 70 of Machine Learning Foundations14m
Exercise: Compute ∂V/∂l for a cylinder with r = 3 and l = 5 where V = π r^2 l
Video class: Advanced Partial-Derivative Exercises — Topic 71 of Machine Learning Foundations02m
Exercise: What is ∂A/∂r for A(r,h) = 2πr^2 + 2πrh?
Video class: Partial Derivative Notation — Topic 72 of Machine Learning Foundations02m
Exercise: Which notation best emphasizes the operator when taking the partial derivative of f(x, y) with respect to x
Video class: The Chain Rule for Partial Derivatives — Topic 73 of Machine Learning Foundations09m
Exercise: Given y = f(u, v) with u = g(x, z) and v = h(x, z), what is the correct expression for the partial derivative ∂y/∂x?
Video class: Exercises on the Multivariate Chain Rule — Topic 74 of Machine Learning Foundations01m
Exercise: Using the multivariate chain rule for y = f(u, v) with u = g(x1, x2) and v = h(x1, x2), what is ∂y/∂x1?
Video class: Linear Regression Fit Point by Point — Topic 75 of Machine Learning Foundations15m
Exercise: In the four-step loop for single-point linear regression y = m x + b, which step computes the gradient ∂C/∂m and ∂C/∂b?
Video class: The Gradient of Quadratic Cost — Topic 76 of Machine Learning Foundations15m
Exercise: Compute the partial derivative dC/dm for C = (y_hat - y)^2 with y_hat = m x + b
Video class: Gradient Descent (Hands-on with PyTorch) — Topic 77 of Machine Learning Foundations12m
Exercise: Which update direction reduces cost in multivariate gradient descent?
Video class: The Gradient of Mean Squared Error — Topic 78 of Machine Learning Foundations24m
Exercise: Gradient of mean squared error with respect to m in linear regression y_hat = m x + b
Video class: Backpropagation — Topic 79 of Machine Learning Foundations06m
Exercise: Which description best defines backpropagation during neural network training?
Video class: Higher-Order Partial Derivatives — Topic 80 of Machine Learning Foundations12m
Exercise: Compute the mixed second-order partial derivative for z = x^2 + 5xy + 2y^2
Video class: Exercise on Higher-Order Partial Derivatives — Topic 81 of Machine Learning Foundations01m
Exercise: For z = x^3 + 2xy, which set lists (z_xx, z_yy, z_xy) correctly?
Video class: Integral Calculus — The Final Segment of Calculus Videos in my ML Foundations Series02m
Exercise: Which task in binary classification directly applies integral calculus?
Video class: Binary Classification — Topic 82 of Machine Learning Foundations09m
Exercise: Which evaluation metric better captures model performance for binary classification than a single 0.5 threshold when outputs reflect varying confidence levels
Video class: The Confusion Matrix — Topic 83 of Machine Learning Foundations02m
Exercise: Identify the false negative in a binary confusion matrix
Video class: The ROC Curve (Receiver-Operating Characteristic Curve) — Topic 84 of Machine Learning Foundations10m
Exercise: Given a ROC curve that passes through the points FPR,TPR = 0.5,1.0 and 0.5,0.5 and 0,0.5 with the standard endpoints 0,0 and 1,1 added, what is the approximate area under the curve AUC
Video class: What Integral Calculus Is — Topic 85 of Machine Learning Foundations06m
Exercise: What is the primary use of integral calculus in machine learning?
Video class: The Integral Calculus Rules — Topic 86 of Machine Learning Foundations05m
Exercise: Compute the indefinite integral of x^4 + 9x^2 using standard integration rules
Video class: Indefinite Integral Exercises — Topic 87 of Machine Learning Foundations01m
Exercise: Compute the indefinite integral ∫(12x^5 - x) dx
Video class: Definite Integrals — Topic 88 of Machine Learning Foundations07m
Exercise: Evaluate the definite integral of 0.5 x dx from x = 1 to x = 2
Video class: Numeric Integration with Python — Topic 89 of Machine Learning Foundations04m
Exercise: What is the value of the definite integral of x/2 over the interval [1, 2] computed numerically?
Video class: Definite Integral Exercise — Topic 90 of Machine Learning Foundations01m
Exercise: Compute the definite integral of 2x over the interval [3, 4]
Video class: Finding the Area Under the ROC Curve — Topic 91 of Machine Learning Foundations03m
Exercise: Given ROC points (0,0), (0,0.5), (0.5,0.5), (0.5,1), (1,1), what is the AUC computed via the trapezoidal rule?
Video class: My Favorite Calculus Resources — Topic 92 of Machine Learning Foundations04m
Exercise: Best calculus approach to compute AUC from an ROC curve with discrete points
