Mathematics & algebra calculator
Partial Derivative Calculator
Compute first-order, second-order, mixed, higher-order, chain rule, and implicit partial derivatives with step-by-step solutions and optional evaluation at a point.
Partial derivative result
Partial Derivative Calculator results
Your partial derivatives will appear here
Enter your function or equation and click Calculate. Results are not shown before you run the calculator.
Requested derivative
Symbolic results are generated with a browser-based math engine. Always review syntax, assumptions, and the simplified expression for coursework or formal use.
Calculator overview
Quick Partial Derivative Calculator Overview
Use this partial derivative calculator to differentiate multivariable functions with respect to one variable at a time. It supports step-by-step work for calculus problems involving x, y, z, or custom variables.
Enter a multivariable function and choose the variable to calculate the partial derivative.
Guide
Partial Derivative Calculator Guide
Use this guide to understand standard partial derivatives, mixed partials, chain rule workflows, implicit partial differentiation, and point evaluation without turning the page into a full calculus textbook.
What This Calculator Does
This partial derivative calculator computes symbolic partial derivatives of multivariable functions. It can find first-order partials, all four second-order partial derivatives for two-variable functions, mixed partial derivatives, higher-order requests, chain rule partial derivatives, and implicit partial derivatives.
The calculator is designed as a symbolic multivariable derivative calculator with steps. It also supports point evaluation after the symbolic derivative has been calculated.
How Partial Derivatives Work
A partial derivative measures how a function changes with respect to one variable while the other variables are treated as constants.
Partial derivative idea
fx = partial f / partial x, treating y, z, ... as constants fThe multivariable function you enter
xThe active variable for this derivative
y,zOther variables held constant during that step
fxThe first partial derivative with respect to x
First and Second Partial Derivatives
First partial derivatives include fx and fy. Second partial derivatives repeat the process, producing fxx, fyy, fxy, and fyx for a two-variable function.
First x partial
Differentiate f once with respect to x.
First y partial
Differentiate f once with respect to y.
Second x partial
Differentiate fx with respect to x again.
Mixed second partial
Differentiate fx with respect to y.
Mixed second partial
Differentiate fy with respect to x.
Second y partial
Differentiate fy with respect to y again.
Mixed Partial Derivatives
Mixed partial derivatives use more than one active variable, such as fxy, fyx, fxyz, or fxxy. The order string tells the calculator which variable to differentiate by at each step.
In many classroom examples, fxy and fyx are equal when the function is sufficiently smooth. The calculator still computes both when you request mixed second partials so you can compare them directly.
Chain Rule for Partial Derivatives
Chain rule partial derivative problems appear when an outer function depends on intermediate variables that depend on other variables. For example, z = f(x,y), x = x(s,t), and y = y(s,t).
Two-variable chain rule
dz/ds = (partial f / partial x)(dx/ds) + (partial f / partial y)(dy/ds) Chain Rule mode builds this expansion, substitutes the intermediate definitions, and simplifies the final derivative.
Implicit Partial Differentiation
Implicit partial differentiation is useful when a relationship is given as an equation instead of an explicit function. For F(x,y,z)=0, the common formulas are based on differentiating both sides and solving for the desired derivative.
Implicit partial formula
partial z / partial x = -F_x / F_z The same idea works for partial z over partial y by replacing F_x with F_y, as long as F_z is not zero at the point of interest.
Evaluation at a Point
Point evaluation happens after the symbolic derivative is found. First calculate the derivative expression, then substitute values such as x = 1, y = 2, and z = 3 to get a numeric result.
This is useful when a homework problem asks for a partial derivative at a point or when you want to check the rate of change at one specific coordinate.
How to Use
- 1Choose the mode
Use standard, chain rule, or implicit partial derivative mode.
- 2Enter the function or equation
Use standard math syntax such as x^2*y, sin(x*y), exp(x), or log(x).
- 3Choose the derivative
Select a preset or enter an order string such as x, xy, yx, or xxy.
- 4Add a point if needed
Turn on point evaluation and enter the coordinate values.
- 5Click Calculate
Review the symbolic result, simplified result, and step-by-step solution.
Tips / Notes
- Simplify after differentiating when possible, especially for products, powers, exponentials, and trig functions.
- Chain rule mode is best for compositions.
- Implicit mode is best when the function is defined by an equation rather than solved explicitly.
- Mixed partials may match under common smoothness conditions, but it is still useful to compute both.
- For point evaluation, check the symbolic result first so you know what is being substituted.
- Choose the active variable carefully when the expression has more than one symbol.
FAQ
Frequently Asked Questions
Clear answers about first partials, second partials, mixed derivatives, chain rule, implicit forms, and point values.
What does the Partial Derivative Calculator do?
It computes symbolic partial derivatives for multivariable functions, including first-order, second-order, mixed, higher-order, chain rule, and implicit partial derivatives.
How do I find the first partial derivatives of a function?
Choose Standard Partial Derivatives, enter the function, select the first-order preset, and click Calculate. The calculator returns fx, fy, fz, and any other detected first partials.
How do I calculate all four second-order partial derivatives?
Use the all four second-order partials preset. For a two-variable function, the calculator returns fxx, fxy, fyx, and fyy with concise steps.
What is a mixed partial derivative?
A mixed partial differentiates with respect to more than one variable, such as fxy or fyx. For sufficiently smooth functions, these are often equal.
Can this calculator do chain rule partial derivatives?
Yes. Chain Rule mode lets you enter an outer function and intermediate variable definitions, then calculates the requested derivative such as dz/ds or dz/dt.
Can this calculator find implicit partial derivatives and evaluate them at a point?
Yes. Implicit mode rewrites an equation as F=0, solves expressions such as partial z over partial x, and can evaluate the symbolic result after you enter point coordinates.