Optimization Calculators

Tools for calculus and optimization, essential for machine learning and deep learning.

Gradient Calculator

Function f(x, y, ...)

Point (x, y, ...)

Partial Derivative Calculator

Function f(x, y, ...)

Variable to Differentiate With Respect To

Hessian Matrix Calculator

Calculate the Hessian matrix of second-order partial derivatives for a scalar function.

Function (e.g., x^2*y + y^3)

Variables (comma-separated, order matters, e.g., x, y)

Evaluation Points for Variables (optional, e.g., 1, 2)

Jacobian Matrix Calculator

Vector-valued function F (comma-separated)

Variables (comma-separated)

Gradient Descent Update Rule

Calculates one step of gradient descent: θ_new = θ_current - α * ∇J(θ_current)

Current Parameter (θ_current)

Gradient (∇J(θ_current))

Learning Rate (α)

Chain Rule Calculator

Calculate ∂z/∂x for z = f(y₁, y₂, ...), where each yᵢ = gᵢ(x₁, x₂, ...).

Outer function z = f(y₁, y₂, ...)

Intermediate Variables in Outer Function (comma-separated, e.g., y1, y2)

Inner Functions yᵢ = gᵢ(x₁, x₂, ...)

Independent Variables (x₁, x₂, ... in inner functions, comma-separated)

Target Independent Variable (for ∂z/∂x)

Evaluation Points for Independent Variables (optional, comma-separated)

Lagrange Multiplier Calculator

Objective function f(x, y)

Constraint function g(x, y) = 0

Backpropagation Calculator (1 Epoch)

Demonstrates forward and backward pass for a 2-input, 2-hidden-neuron, 1-output-neuron network using sigmoid activation and MSE loss.

Inputs:

Input x1:

Input x2:

Target Output (y_target):

Weights & Biases:

Hidden Neuron h1:

w_x1_h1 (x1 to h1):

w_x2_h1 (x2 to h1):

Bias b_h1:

Hidden Neuron h2:

w_x1_h2 (x1 to h2):

w_x2_h2 (x2 to h2):

Bias b_h2:

Output Neuron o1:

w_h1_o1 (h1 to o1):

w_h2_o1 (h2 to o1):

Bias b_o1: