How to read the Partial Derivative calculation
A partial derivative varies one coordinate while freezing the others, producing a directional coordinate rate.
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.
Hand-checking the Partial Derivative result
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.
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.
Multivariable optimization, surfaces, fields, economics, and differential equations use partial derivatives. A related application of Partial Derivative is implicit slope.
A second verification of Partial Derivative
Inspect the domain of Function f(x,y) before using Partial Derivative. Keep exact Partial Derivative work separate from x coordinate.
Test Partial Derivative on a constant or linear function. Refine any numerical Partial Derivative step and compare the approximation.
Preserve Function f(x,y) when checking the Partial Derivative output. Keep x coordinate under the same convention and estimate Partial derivative. A controlled change to y coordinate should move the Partial Derivative Partial derivative in a mathematically consistent direction.
Validating Partial Derivative
Record the original Function f(x,y) before changing Partial Derivative. Keep x coordinate with that record and attach its own Partial derivative. A later Partial Derivative trial then remains distinguishable from the first calculation.
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.