Partial Derivative Calculator

Look up partial derivative rules for common multivariable functions. Select a function type and enter coefficients to see ∂f/∂x and ∂f/∂y with step-by-step derivation. Covers polynomials, exponentials, logarithms, and trigonometric functions.

Partial Derivative Reference

What Is a Partial Derivative?

A partial derivative measures how a multivariable function changes with respect to one variable while all other variables are held constant. The notation ∂f/∂x (read "partial f partial x") represents the rate of change of f with respect to x, treating y (and any other variables) as constants.

Partial derivatives are foundational in multivariable calculus, appearing in gradient vectors, directional derivatives, the Jacobian matrix, optimization problems (including machine learning), thermodynamics, economics, and engineering analysis.

Partial Derivative Rules

Power Rule: f = x^m · y^n ∂f/∂x = m·x^(m-1) · y^n ∂f/∂y = n·x^m · y^(n-1) Exponential: f = e^(ax+by) ∂f/∂x = a·e^(ax+by) ∂f/∂y = b·e^(ax+by) Logarithm: f = a·ln(x) + b·ln(y) ∂f/∂x = a/x ∂f/∂y = b/y Sine/Cosine: f = sin(ax) · cos(by) ∂f/∂x = a·cos(ax) · cos(by) ∂f/∂y = -b·sin(ax) · sin(by) Chain Rule: f = g(h(x,y)) ∂f/∂x = g'(h) · ∂h/∂x

Second-Order Partial Derivatives

You can differentiate a partial derivative again to get second-order partials. The notation ∂²f/∂x² represents differentiating with respect to x twice. The mixed partial ∂²f/∂x∂y represents differentiating with respect to x first, then y (or vice versa). By Clairaut's theorem, if f has continuous second partials, ∂²f/∂x∂y = ∂²f/∂y∂x.

Applications

FieldApplication
Machine LearningGradient descent uses partial derivatives of the loss function
PhysicsMaxwell's equations, heat equation, wave equation
EconomicsMarginal utility, optimization of profit functions
EngineeringStress analysis, fluid dynamics
FinanceBlack-Scholes option pricing (Greeks: Delta, Gamma, Theta)

Frequently Asked Questions

What is the difference between ∂ and d?+
The symbol d (ordinary derivative) is used for single-variable functions. The symbol ∂ (partial derivative) is used for multivariable functions and means we differentiate with respect to one variable while holding all others constant.
How do you find ∂f/∂x?+
Treat y (and all other variables) as constants and differentiate normally with respect to x using standard calculus rules (power rule, chain rule, product rule, etc.).
What is the gradient vector?+
The gradient ∇f is the vector of all partial derivatives: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z, ...). It points in the direction of steepest ascent of the function and is fundamental in optimization and machine learning.

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