Partial derivatives
Objective. Compute and interpret first and higher partial derivatives.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A partial derivative measures the rate of change of a multivariable function with respect to one variable while all other variables are held constant, written ∂f/∂x or fₓ. To compute ∂f/∂x, treat y (and any other variables) as constants and differentiate using ordinary single-variable rules. Higher partial derivatives, like ∂²f/∂x² or the mixed partial ∂²f/∂x∂y, come from differentiating again, and Clairaut's theorem states that for sufficiently smooth functions, mixed partials are equal regardless of order: fₓᵧ = fᵧₓ. For f(x, y) = x²y³, we get fₓ = 2xy³ and fᵧ = 3x²y², so fₓᵧ = 6xy² which matches fᵧₓ.
Worked example
Find the partial derivative ∂f/∂y of f(x, y) = 3x²y⁴ + 5xy.
- 1Treat x as a constant and differentiate each term with respect to y.
- 2Differentiate 3x²y⁴: the coefficient 3x² stays, and y⁴ becomes 4y³, giving 12x²y³.
- 3Differentiate 5xy with respect to y: 5x stays, y becomes 1, giving 5x.
- 4Add the two results together for the full partial derivative.
Answer: ∂f/∂y = 12x²y³ + 5x
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Practice
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Start practisingCommon mistake. A common slip is differentiating with respect to x but forgetting that y is treated as a constant, accidentally also differentiating any y terms; keep non-target variables completely fixed throughout each partial derivative.
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