∂
Guide
What Are Partial Derivatives? A Complete Beginner's Guide
From the limit definition to real-world applications in physics and machine learning — everything you need to get started.
Enter any multivariable function, pick your variable, and get the exact partial derivative — with step-by-step explanations. Free, instant, no signup.
Quick Examples
Click any example to populate the calculator and see results instantly.
Calculus Reference
A partial derivative measures how a multivariable function changes as one variable changes, while all other variables are held constant. It extends single-variable differentiation to functions of several variables.
Second-order partial derivatives describe the curvature of a function's surface. They appear in the Hessian matrix and are essential for optimization and classifying critical points.
Mixed partials differentiate with respect to two different variables. Schwarz's theorem states that ∂²f/∂x∂y = ∂²f/∂y∂x when both are continuous — the order doesn't matter.
The gradient ∇f collects all partial derivatives into a vector pointing in the direction of steepest ascent. It's foundational in machine learning, physics, and optimization.
Learning Resources
Deepen your understanding of multivariable calculus with our in-depth guides.
From the limit definition to real-world applications in physics and machine learning — everything you need to get started.
Why does the order of differentiation not matter? We break down the theorem and work through detailed examples.
How second-order partial derivatives form the Hessian and determine whether a critical point is a min, max, or saddle point.
FAQ
You can enter any standard mathematical function including polynomials (x^2 + y^2), trigonometric functions (sin(x), cos(y), tan(x*y)), exponentials (exp(x)), logarithms (ln(x)), and combinations of these. Use * for multiplication and ^ for exponents.
Enable the "Mixed Partial" checkbox, then select two (or more) variables by clicking their chips. The calculator will differentiate with respect to each selected variable in sequence, computing ∂²f/∂x∂y or higher-order mixed partials.
A partial derivative (∂f/∂x) measures change in f with respect to x while all other variables are held constant. A total derivative accounts for how all variables change simultaneously and is expressed as df = (∂f/∂x)dx + (∂f/∂y)dy + … The total derivative is used when variables are related (e.g., via a path or constraint).
Yes, completely free. No signup, no account required. All computations run directly in your browser — we don't send your functions to any server.
Partial derivatives appear throughout science and engineering: gradient descent in machine learning uses them to minimize loss functions; thermodynamics uses them to relate pressure, volume, and temperature; economics uses them for marginal analysis; and physics uses them in the equations of electromagnetism (Maxwell's equations) and quantum mechanics (Schrödinger equation).