Partial Derivatives: Rules and Examples Fully Explained

A partial derivative measures how a multivariable function changes when exactly one variable moves and all the others are held frozen. For f(x, y), the partial derivative ∂f/∂x treats y as a constant and differentiates with respect to x using the ordinary rules.

This matters because real systems change one factor at a time, like raising a price while advertising stays fixed. Partial derivatives isolate each influence and underpin gradients, optimisation, and the equations of physics.

Key Concepts

• Notation: ∂f/∂x and fx both mean the partial derivative with respect to x; ∂f/∂y and fy with respect to y. The curly ∂ distinguishes partial from ordinary d.
• Frozen variables are constants: differentiating 3x²y with respect to x treats y like the number 5, giving 6xy; differentiating it with respect to y treats x² as constant, giving 3x².
• Second partials: fxx = ∂²f/∂x² differentiates twice in x; mixed partials like fxy = ∂²f/∂y∂x differentiate first in x, then in y.
• Clairaut's theorem: for functions with continuous second partials, the mixed partials are equal: fxy = fyx. This is a free error check on every computation.
• Geometric meaning: fx(a, b) is the slope of the surface in the x-direction at (a, b); fy is the slope in the y-direction.

Formulas

• Power rule with frozen variables: ∂/∂x (xⁿyᵐ) = n·x^(n−1)yᵐ, since yᵐ is constant in x.
• Terms without x vanish: ∂/∂x of any expression in y alone is 0.
• Clairaut's theorem: ∂²f/∂y∂x = ∂²f/∂x∂y wherever the second partials are continuous.

Worked Example

For f(x, y) = 3x²y + y³ − 2x, find fx, fy, and verify Clairaut's theorem.

1. Partial with respect to x (y frozen): ∂/∂x(3x²y) = 6xy; ∂/∂x(y³) = 0 since y³ is constant in x; ∂/∂x(−2x) = −2. So fx = 6xy − 2. At (1, 2): fx(1, 2) = 6·1·2 − 2 = 12 − 2 = 10.
2. Partial with respect to y (x frozen): ∂/∂y(3x²y) = 3x²; ∂/∂y(y³) = 3y²; ∂/∂y(−2x) = 0. So fy = 3x² + 3y². At (1, 2): fy(1, 2) = 3·1 + 3·4 = 3 + 12 = 15.
3. Mixed partial fxy: differentiate fx = 6xy − 2 with respect to y: ∂/∂y(6xy) = 6x and ∂/∂y(−2) = 0, giving fxy = 6x.
4. Mixed partial fyx: differentiate fy = 3x² + 3y² with respect to x: ∂/∂x(3x²) = 6x and ∂/∂x(3y²) = 0, giving fyx = 6x.
5. Verify Clairaut: fxy = 6x = fyx. They match, confirming the arithmetic.

Common Misconceptions

• Differentiating the frozen variable's terms. When computing ∂/∂x, any term with no x is a constant with derivative zero; "differentiating" y³ with respect to x to get 3y² is wrong.
• Confusing partial with total change. ∂f/∂x measures change along x alone; if y also depends on x, the total derivative needs the chain rule as well.
• Assuming mixed partials always match. Clairaut's theorem needs continuity of the second partials; exotic counterexamples exist, but for every standard function the mixed partials agree.

Real-World Applications

• Economics: marginal productivity is the partial derivative of output with respect to labour or capital, holding the other fixed.
• Physics: the heat equation and wave equation are built from second partial derivatives in space and time.
• Optimisation: finding peaks and valleys of multivariable functions starts by setting all partial derivatives to zero.
• Machine learning: gradient descent updates each parameter using its partial derivative of the loss function.

Partial derivatives use the ordinary rules while all other variables stand still. Compute fx and fy term by term, check the mixed partials with Clairaut's theorem, and you can measure change in any direction.