Calculus

Compute an average rate of change over an interval.

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

Determine a one-sided limit from a piecewise graph.

Evaluate a finite limit using valid algebraic limit laws.

Form the difference quotient for a scalar function.

Compute a simple derivative from a difference-quotient limit.

Diagnose failure of differentiability from unequal one-sided slopes.

Construct a tangent-line equation from a point and derivative.

Approximate a nearby function value using a local linearization.

Differentiate a constant scalar function.

Differentiate a scalar power function on its real domain.

Differentiate a sum of differentiable scalar functions.

Differentiate a constant multiple of a differentiable function.

Differentiate a product of two scalar functions.

Differentiate a composition of two scalar functions.

Differentiate an exponential function with a constant base.

Differentiate a natural logarithm on its real domain.

Compute the second derivative of a scalar function.

Evaluate a scalar function at a vector input.

Identify inputs sharing the same scalar output.

Compute a partial derivative with other coordinates held fixed.

Assemble a gradient from partial derivatives in coordinate order.

Compute a directional rate using a unit direction.

Estimate a scalar change from a small vector perturbation.

Construct the Jacobian of a vector-valued map.

Compute the derivative of a composed vector map using Jacobians.

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

Evaluate a small computational graph in dependency order.

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

Propagate a scalar-output sensitivity backward through a chain.

Accumulate sensitivities at an input used by multiple graph branches.

Propagate a supplied output sensitivity through a vector operation.

Estimate a derivative with a centered finite difference.

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

Compute the gradient of aᵀx with respect to x for a constant real vector a.

Compute the gradient of xᵀAx as (A + Aᵀ)x for a constant real square matrix A.

Compute the gradient of one-half the squared Euclidean residual norm ||Ax − b||² as Aᵀ(Ax − b) for fixed A and b.

Assemble the gradient of a scalar function of a real matrix from its entrywise partial derivatives.

Compute ∇_Z [||Z − Y||_F²/(2m)] = (Z − Y)/m for fixed real Y and a fixed positive scalar m.

Estimate the first-order change of a scalar objective under a small matrix perturbation using the Frobenius inner product of its gradient and that perturbation.

Compute the left-input gradient GBᵀ for Z = AB from a supplied incoming scalar-loss gradient G.

Compute the right-input gradient AᵀG for Z = AB from a supplied incoming scalar-loss gradient G.