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Partial-derivative meaning

Why Are Other Variables Constant in Partial Derivatives?

Treating y as constant in a partial derivative with respect to x does not mean y is always a universal constant. It means this one comparison changes x while keeping the other coordinate fixed.

Direct answer

A partial derivative measures change in one coordinate direction, so the other independent variables are held fixed during that measurement.

This interpretation assumes the listed variables are independent coordinates; variables connected by a constraint require a different analysis.

Directional argument: change one coordinate

For a function f(x,y), the partial derivative with respect to x compares f(x+h,y) with f(x,y). The y-coordinate is identical in both values, so only x changes.

In x^2y, the fixed y acts like a coefficient. Differentiating with respect to x gives 2xy. In y^3, no x changes at all, so the derivative of that entire term is zero.

For sin(xy), y remains fixed while the inner expression xy changes at rate y. The chain rule therefore gives y cos(xy).

The held-constant instruction belongs to the selected task. Switching to a partial derivative with respect to y reverses the roles of the variables.

1
f_x(x,y) = lim(h->0) [f(x+h,y)-f(x,y)]/h
2
d/dx (x^2y) = 2xy when y is fixed
3
d/dx (y^3) = 0 when y is fixed

Worked examples

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

The y^3 term is constant with respect to x and becomes zero.

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

Hold y fixed, then apply the chain rule to the inner product xy.

partial/partial y exp(x^2+y^2)
2y exp(x^2+y^2)

The x^2 portion is fixed, while y^2 contributes the inner derivative 2y.

Common mistakes

  • • Differentiating every visible variable even when only one active variable was selected.
  • • Deleting a held variable that should remain as a coefficient.
  • • Calling a held variable globally constant rather than constant for this partial-derivative task.
  • • Applying the held-constant rule when the variables are linked by an explicit constraint.

Try the worked examples in the calculator

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

偏微分計算機:x2 y+y3x^{2}\,y + y^{3}偏微分計算機:sin⁡(x y)\sin\left(x\,y\right)偏微分計算機:x3 y2x^{3}\,y^{2}
Open the calculator without a preset: 偏微分計算機

References

Frequently asked questions

Does holding y constant mean y equals zero?+

No. Its value stays fixed but can be any allowed number, so it often remains as a coefficient in the derivative.

What if y depends on x?+

Then an ordinary or total derivative may be required. The standard partial derivative treats the coordinates as independently adjustable.

Can a term containing y survive an x partial derivative?+

Yes. A term such as x^2y changes with x, so y remains in the result as a fixed multiplier.

Conclusion

Hold the non-selected coordinates fixed, not necessarily at zero. Then ask which terms still change with the active variable and differentiate only those changes.

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