Partial Derivative Calculator
Enter a multivariable function and the variable to differentiate with respect to — the others are held constant.
Use ^ for powers and * or implied for multiply, e.g. x^2*y^3.
A partial derivative measures how a multivariable function changes as one variable varies while the others are held constant. To find it, differentiate with respect to that variable and treat every other variable as a constant. The partial derivative of x²y³ with respect to x is 2xy³.
How to Use This Calculator
Enter your function of two or more variables using ^ for exponents, and enter the single variable you want to differentiate with respect to. The calculator returns the partial derivative, holding every other variable fixed.
Partial derivatives are the foundation of multivariable calculus, used in gradients, optimization and the equations of physics and economics.
How the Calculation Works
Taking a partial derivative is exactly like an ordinary derivative, with one rule: every variable except the one you are differentiating by is treated as a constant number.
So in x²y³, differentiating with respect to x treats y³ as a constant multiplier, giving 2x·y³. Differentiating the same function with respect to y instead would treat x² as constant, giving x²·3y² = 3x²y².
∂f/∂x treats y, z, … as constants and differentiates in x
Worked Example
For f = x²y³, the partial derivative with respect to x is found by treating y³ as a constant: the derivative of x² is 2x, so ∂f/∂x = 2xy³.
With respect to y, x² is held constant and the derivative of y³ is 3y², giving ∂f/∂y = 3x²y².
Assumptions and Limitations
- Differentiates with respect to the single variable you name
- All other variables are treated as constants
- Enter standard functions (sin, cos, exp, ln) and ^ for powers
- Returns a first-order partial derivative
Who Uses a Partial Derivative Calculator
Multivariable calculus students and engineers use this to differentiate a function with respect to one variable while holding the others constant. Run it to check homework, build gradients, or work through optimization problems. Enter the function and the variable to get the partial derivative.
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Open calculator →Frequently Asked Questions
What is a partial derivative?
It is the derivative of a multivariable function with respect to one variable, holding the others constant. It measures how the function changes as just that one input varies.
How is a partial derivative different from an ordinary derivative?
The method is the same, but a partial derivative treats all other variables as constants. An ordinary derivative applies to single-variable functions where there is nothing else to hold fixed.
What is ∂f/∂x of x²y³?
It is 2xy³. Treating y³ as a constant, the derivative of x² is 2x, so the result is 2x times y³.
Can I take the partial derivative with respect to y?
Yes — just enter y as the variable. For x²y³, that gives 3x²y², holding x² constant and differentiating y³.
Does this do second-order partials?
This tool returns the first partial derivative. To get a second-order partial, run the tool again on the result with respect to the desired variable.
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