Partial Derivative Calculator
How to find a partial derivative
Enter a polynomial in x, y and z, pick the variable, and press Calculate. The calculator treats the other variables as constants and applies the power rule term by term: d/dx[x^n] = n·x^(n−1).
Worked example
f = x²y + 3xy². ∂f/∂x = 2xy + 3y² (y held constant). ∂f/∂y = x² + 6xy (x held constant).
What is supported
Polynomial terms only (like 2x^2y, 3xy, z, 5). Terms without the chosen variable differentiate to 0. Trigonometric or exponential functions are not supported yet, the calculator says so rather than guessing.
See the method applied in our guide: partial derivatives of x²y³ + e^(xy), with both variables worked.
Frequently asked questions
The rate of change of a multivariable function when one variable changes and the others stay fixed. ∂f/∂x measures how f changes as x moves, with y and z frozen.
If a term has no x in it, its partial derivative with respect to x is 0, it is constant as far as x is concerned.
A normal derivative is for one-variable functions. A partial derivative is the same idea applied to one variable at a time in a multivariable function.
What holding a variable constant really means
A partial derivative slices a surface. Fix y and the surface becomes an ordinary curve in x, whose slope is the partial derivative with respect to x. Powers and sums then behave as in one variable work. The only new discipline is bookkeeping: every other symbol is a fixed number throughout, so 4xy differentiated with respect to x gives 4y, not 4.
Notice that y never moves here. The same pattern defines the other partial derivatives: change one input by h, keep the rest fixed, and measure how fast the output responds.
Example: Find both partial derivatives of f = x^3 y^2 + 4xy + y^3 at (2, 1)
Step 1, Differentiate with respect to x: Hold y constant. fx = 3x^2 y^2 + 4y. The y^3 term has no x, so it contributes 0.
Step 2, Differentiate with respect to y: Hold x constant. fy = 2x^3 y + 4x + 3y^2.
Step 3, Evaluate fx at (2, 1): fx = 3(4)(1) + 4(1) = 16.
Step 4, Evaluate fy at (2, 1): fy = 2(8)(1) + 4(2) + 3(1) = 27. At this point f rises faster in the y direction than in the x direction.
When partial derivatives are used
Use them whenever an output depends on several inputs and you need the effect of one alone: how a price change affects demand while costs stay fixed, or which input a model output is most sensitive to. They also build gradients and optimisation.
Common mistakes
- Dropping the held variable: The derivative of 4xy with respect to x is 4y. Treating y as 1 loses it.
- Deleting terms too eagerly: A term only vanishes if it contains none of your chosen variable. x^3 y^2 survives in both fx and fy.
- Evaluating too soon: Differentiate symbolically first, then substitute. Substituting first turns the function into a number.
- Mixing up the variable: State clearly whether an answer is fx or fy.
More questions
It is a second partial derivative taken in two different variables, for example first with respect to x and then with respect to y. For smooth functions the order does not matter, so fxy and fyx are equal.
It is the slope of the tangent line to a slice of the surface. Slice by holding y fixed, and fx gives the slope of that slice in the x direction at the point.
Yes. Differentiate the first partial derivative again, with respect to the same or a different variable. Second partial derivatives describe curvature and are used in tests for maxima and minima.