Holds other variables constant
While finding ∂/∂x, symbols such as y and z are treated as constants. Thus ∂(x²y)/∂x = 2xy.
Partial Derivative Calculator with Steps
Enter a multivariable function and choose an order of differentiation. The calculator finds exact first, higher-order, or mixed partial derivatives, explains each rule, and can evaluate the result at a point—all locally in your browser.
Enter a multivariable function
Use ^ for powers and * between variables: write x*y, not xy. Functions include sin, cos, tan, exp, ln, log, sqrt, and abs.
Examples: x, x,x, or x,y. Maximum order: 4.
Assign every variable used in the final result.
Try an example
Keyboard shortcut: Ctrl/⌘ + Enter.
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Partial derivative
Original function
Enter a function to begin.
Result
The exact partial derivative will appear here.
Step-by-step partial differentiation
Rule-by-rule working will appear here.
Calculator methodology and limits
- Developed and reviewed by
- Starlight Robotics calculator team
- Last reviewed
- Method
- Parse an expression tree → hold non-selected variables constant → apply symbolic differentiation rules → simplify after each stage.
- Input limits
- Single-letter variables, 300-character functions, and differentiation order up to 4.
- Privacy
- Your function and point values stay in your browser; no solving API receives them.
- Domain note
- Returned formulas may require restrictions from denominators, logarithms, roots, inverse trig functions, or absolute values.
How to use the partial derivative calculator
- Enter a function such as
x^2*y + sin(x*y). Use an asterisk between adjacent variables. - Enter variables in the order they should be applied. Use
xfor ∂/∂x,x,xfor ∂²/∂x², orx,yto differentiate first by x and then by y. - Optionally enter a point such as
x=1, y=2to evaluate the final derivative. - Select Calculate partial derivative, then review, copy, or download the steps.
What the calculator does
Supports mixed partials
A sequence such as x,y applies ∂/∂x first and ∂/∂y second, producing the operator ∂²/(∂y∂x).
Applies standard rules
Linearity, product, quotient, power, chain, trigonometric, exponential, logarithmic, root, and absolute-value rules are supported.
Checks point values
When every remaining variable is assigned a finite value, the calculator evaluates the final symbolic partial and reports undefined or non-real results clearly.
Worked partial derivative examples
First partial
f(x,y)=x²y+sin(xy)
Holding y constant gives ∂f/∂x=2xy+y cos(xy).
Pure second partial
f(x,y)=x³y²
Differentiate twice by x: ∂²f/∂x²=6xy².
Mixed partial
f(x,y)=x²y
First ∂f/∂x=2xy; then ∂/∂y gives ∂²f/(∂y∂x)=2x.
Partial derivative calculator FAQs
What is a partial derivative?
It is the derivative of a multivariable function with respect to one variable while the other variables are held constant. It measures change in one coordinate direction.
Can this calculator find mixed partial derivatives?
Yes. Enter variables in application order. For x,y, the tool differentiates first with respect to x and then with respect to y.
Why must I write x*y instead of xy?
The parser reserves multi-letter names for functions. An asterisk makes the two distinct variables unambiguous. Numeric multiplication such as 2x remains supported.
Does the order of mixed partial derivatives matter?
It can. If the relevant second partials are continuous near a point, Clairaut’s theorem allows the order to be exchanged there. At singularities or for less-smooth functions, calculate both orders separately.
Can I use parameters?
Yes. Any single-letter symbol not selected at a differentiation stage is treated as a constant at that stage. For example, differentiating a*x^2 by x gives 2*a*x.
Are trigonometric inputs measured in radians?
Yes. Standard calculus identities such as ∂sin(xy)/∂x = y cos(xy) assume radian measure.
Does my expression leave the browser?
No. Symbolic parsing, differentiation, simplification, and evaluation run on your device.