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Glacius

Builds on functions

Calculus

Explore change, gradients, and the chain rule that makes learning algorithms possible.

Start with average rate of change 29 lessons · 8 units
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Explore gradients

Inside this subject

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01Limits and continuity4 lessons
  1. 01
    Average rate of change

    Compute an average rate of change over an interval.

  2. 02
    Approaching a limit

    Infer a finite limit from nearby values while distinguishing the endpoint value.

  3. 03
    One-sided limits

    Determine a one-sided limit from a piecewise graph.

  4. 04
    Limit algebra

    Evaluate a finite limit using valid algebraic limit laws.

02The derivative3 lessons
  1. 01
    Difference quotients

    Form the difference quotient for a scalar function.

  2. 02
    Derivative definition

    Compute a simple derivative from a difference-quotient limit.

  3. 03
    Corners and cusps

    Diagnose failure of differentiability from unequal one-sided slopes.

03Algebraic derivative rules6 lessons
  1. 01
    Constant rule

    Differentiate a constant scalar function.

  2. 02
    Power rule

    Differentiate a scalar power function on its real domain.

  3. 03
    Sum rule

    Differentiate a sum of differentiable scalar functions.

  4. 04
    Constant multiple rule

    Differentiate a constant multiple of a differentiable function.

  5. 05
    Product rule

    Differentiate a product of two scalar functions.

  6. 06
    Scalar chain rule

    Differentiate a composition of two scalar functions.

04Shape and approximation1 lessons
  1. 01
    Second derivatives

    Compute the second derivative of a scalar function.

05Functions of several variables2 lessons
  1. 01
    Multivariable functions

    Evaluate a scalar function at a vector input.

  2. 02
    Level sets

    Identify inputs sharing the same scalar output.

06Partial derivatives and gradients3 lessons
  1. 01
    Partial derivatives

    Compute a partial derivative with other coordinates held fixed.

  2. 02
    Gradients

    Assemble a gradient from partial derivatives in coordinate order.

  3. 03
    Directional derivatives

    Compute a directional rate using a unit direction.

07Vector derivatives and curvature2 lessons
  1. 01
    Jacobians

    Construct the Jacobian of a vector-valued map.

  2. 02
    Multivariable chain rule

    Compute the derivative of a composed vector map using Jacobians.

08Automatic differentiation8 lessons
  1. 01
    Computational graphs

    Represent a scalar expression as a directed graph of elementary operations.

  2. 02
    Forward evaluation

    Evaluate a small computational graph in dependency order.

  3. 03
    Local derivatives

    Compute one graph operation's local derivative with respect to an input.

  4. 04
    Reverse-mode differentiation

    Propagate a scalar-output sensitivity backward through a chain.

  5. 05
    Shared-path gradients

    Accumulate sensitivities at an input used by multiple graph branches.

  6. 06
    Vector–Jacobian products

    Propagate a supplied output sensitivity through a vector operation.

  7. 07
    Finite differences

    Estimate a derivative with a centered finite difference.

  8. 08
    Gradient checks

    Diagnose a mismatch between analytic and numerical derivatives using a stated tolerance.