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Definition and meaning
A partial derivative measures how a multivariable function changes with one selected variable while the other variables are held constant. For f(x,y), the notation ∂f/∂x means differentiate with respect to x and treat y as a constant; ∂f/∂y reverses those roles.
Why the rule works
A function of two variables forms a surface. Fixing y=b creates a one-variable slice f(x,b), and the ordinary derivative of that slice is the partial derivative with respect to x at y=b. This geometric definition explains the ‘hold the other variables constant’ rule without suggesting those variables are globally constant.
A reliable hand method
Choose the differentiation variable before doing algebra. Mark every other symbol as temporarily constant, apply the ordinary sum, product, power, and chain rules, and then restore the multivariable notation. The selector in this tool lets you compare ∂/∂x and ∂/∂y for the same expression.
Example 1: x^2*y
Answer: ∂/∂x=2xy; ∂/∂y=x^2
- For ∂/∂x, y is a constant coefficient.
- For ∂/∂y, x^2 is a constant coefficient.
Example 2: x^2+y^2
Answer: ∂/∂x=2x; ∂/∂y=2y
- Differentiate only terms containing the selected variable.
- The other squared term contributes zero.
Example 3: e^(x*y)
Answer: ∂/∂x=y e^(xy)
- Use the chain rule on the exponential.
- With respect to x, the inner derivative of xy is y.
Example 4: sin(x+y)
Answer: ∂/∂y=cos(x+y)
- Preserve x+y inside cosine.
- The inner derivative with respect to y is 1.
Common mistakes
Do not differentiate every variable at once. The selected variable determines which symbols behave like constants. Avoid replacing a held-constant variable with a number; it remains part of the final formula. Mixed partial notation also records order, although sufficiently smooth functions often have equal mixed partials.
How to verify an answer
Differentiate the expression independently, compare the structure of each term, and substitute a safe numerical value into both the symbolic derivative and a centered difference quotient. Avoid points outside the original domain and points near a discontinuity. A matching numerical value increases confidence, but it cannot establish equality for every input. Algebraic reasoning and domain analysis remain necessary.
When the calculator and handwritten work differ, first check parentheses, multiplication symbols, the selected variable, and any missing chain-rule factor. Then factor or expand both answers to see whether they are equivalent. The DerivCalc methodology explains why simplification is deliberately conservative.
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Questions about this calculator
Can I use the Partial Derivative Calculator for homework checks?
Yes. Compare the result and named rules with your own work, then check domain restrictions. The numerical spot-check is a consistency test, not a formal proof.
Why can an equivalent answer look different?
Factoring, expanding, reciprocal notation, and trigonometric identities can produce different-looking expressions with the same values on their shared domain.
Does the calculator replace learning the rule?
No. It is designed to expose the rule and intermediate reasoning. The worked examples explain how to select and apply the method by hand.
Editorial standard: formulas and notation are checked against OpenStax Calculus Volume 1, Chapter 3. Content is written and maintained by SilverCodeLabs. Report a correction with the exact expression and expected result.