Reading the calculation
A partial derivative varies one coordinate while freezing the others, producing a directional coordinate rate.
A correct partial derivative therefore depends on choosing the model before entering the numbers.
Math calculator
Partial Derivative Calculator
Differentiate a two-variable function with respect to x or y while holding the other fixed. Input changes update both partial derivative and the supporting steps.
Partial Derivative inputs
Complete the fields
Partial Derivative in a worked case
For x²y+sin y at (2,1), the x partial is 2xy=4.
Calculating it by hand
Perturb only the selected coordinate symmetrically and divide the resulting change by the step width.
Reading the Partial Derivative result
Holding the wrong variable fixed changes the question. A partial derivative is not automatically the total derivative along a path.
Carry the available precision through partial derivative, then round the final output rather than its intermediate parts.
Practical meaning
Multivariable optimization, surfaces, fields, economics, and differential equations use partial derivatives. A related application of Partial Derivative is implicit slope.
Checking the model and magnitude
For this partial derivative calculation, the labels Function f(x,y), x coordinate, y coordinate and Differentiate with respect to carry mathematical meaning. A transposed entry can remain numerically valid while describing an entirely different setup.
Switching the selected variable can give a completely different magnitude and unit. A small controlled input change is enough to test the expected direction.
An exact expression can preserve factors, radicals, or π that a decimal hides. Use the representation suited to the next task and label it clearly as partial derivative.
Implicit differentiation combines both partial derivatives to follow a level curve. Checking the requested noun is often enough to select the right model.
Reproducibility here depends on the inputs more than the interface. Preserve Function f(x,y), x coordinate, y coordinate and Differentiate with respect to, the operation shown, and enough unrounded digits for the next calculation.
Reproducing this result later
Keep Function f(x,y), x coordinate, y coordinate, and Differentiate with respect to with the Partial Derivative output. Preserve more digits internally than the final report needs. Subtraction in finite differences and cumulative integration can lose significant digits, so rounding intermediate samples can damage the final estimate disproportionately.
For this partial derivative result, treat the shown digits as conditional on the entered domain and sampling choices. Keeping those choices with the value helps distinguish a stable computation from a coincidental rounded match. Hold the nonselected variable fixed and repeat the estimate with a smaller step before reporting the slope.
Questions about Partial Derivative
What stays fixed?
Every nonselected variable.
Is this a directional derivative?
It is the coordinate-direction case.
Can the partials differ?
Yes.