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Choosing the active variable

Partial Derivative with Respect to x vs. y

The expression does not determine one unique partial derivative until a direction is chosen. Writing the active and held variables beside the problem prevents a common class of otherwise plausible mistakes.

Direct answer

With respect to x, hold y fixed; with respect to y, hold x fixed. The same expression can therefore produce two different valid derivatives.

Always identify the active variable before applying derivative rules, and preserve any other variables that act as coefficients.

Side-by-side method: mark what changes

Start with f(x,y)=x^2y+y^3. For the x partial, y is fixed: x^2y becomes 2xy and y^3 becomes zero.

For the y partial, x is fixed: x^2y becomes x^2 and y^3 becomes 3y^2. The result is x^2+3y^2.

Neither result is more complete than the other. They measure slopes along different coordinate directions on the same surface.

A reliable workflow is to write active: x, held: y or active: y, held: x before differentiating term by term.

1
f(x,y)=x^2y+y^3
2
f_x(x,y)=2xy
3
f_y(x,y)=x^2+3y^2

Worked examples

partial/partial x [x^2y+y^3]
2xy

Only terms that change with x contribute.

partial/partial y [x^2y+y^3]
x2+3 y2x^{2} + 3\,y^{2}

Treat x^2 as a fixed coefficient and differentiate both y terms.

partial/partial x sin(xy)
y cos(xy)

The inner derivative of xy with respect to x is y.

Common mistakes

  • • Using the same answer for both active variables.
  • • Dropping a held variable that should remain as a coefficient.
  • • Differentiating a term that contains only held variables.
  • • Changing the active variable in the calculator without checking the displayed condition.

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

Calculadora de derivadas parciais:x2 y+y3x^{2}\,y + y^{3}Calculadora de derivadas parciais:x2 y+y3x^{2}\,y + y^{3}Calculadora de derivadas parciais:sin⁡(x y)\sin\left(x\,y\right)
Open the calculator without a preset: Calculadora de derivadas parciais

References

Frequently asked questions

Can the x and y partial derivatives be equal?+

They can coincide for a particular function or point, but they are computed with different held-variable conditions.

Which variable does the calculator use by default?+

The active-variable control determines the direction. Check that label before solving or opening a guided example.

What happens to a mixed term such as xy?+

Its x partial is y, while its y partial is x. The held variable remains as a coefficient.

Conclusion

Choose the direction first, state which variables are held fixed, and then process every term with that condition visible. The two partial derivatives answer different questions.

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