Stuck on a multivariable function and not sure how to isolate one variable’s rate of change? This partial differentiation calculator does the heavy lifting. Enter your function, choose the variable to differentiate with respect to, and get the result instantly — no textbook flipping, no formula hunting.

It works for first-order, second-order, and mixed partial derivatives across functions with two or more variables. Whether you’re checking homework or solving an optimization problem, the answer is one click away.

How to Use the Partial Differentiation Calculator

Using the calculator is quick. Here’s what each input means:

• Function — Enter your multivariable function using standard notation. Write x² as x^2, multiplication as x*y, and use sin(x), cos(x), ln(x), and e^x for common functions. Example: x^3 + 3*x*y + y^2
• Differentiation Variable — Choose the variable you want to differentiate with respect to (x, y, z, etc.). All other variables are treated as constants.
• Order — Select 1st order for a standard partial derivative, 2nd order to differentiate twice, or specify mixed (e.g., x then y) for a mixed partial.

Hit Calculate, and the tool returns the partial derivative with a step-by-step breakdown showing which differentiation rules were applied.

Input tip: The calculator is case-sensitive. If your function uses lowercase x, don’t enter uppercase X — especially on mobile where keyboards auto-capitalize.

---

What Is Partial Differentiation?

Partial differentiation is the process of finding how a multivariable function changes when one input variable changes, while all other variables stay fixed.

Think of it this way: imagine you’re standing on a hilly landscape. The height at any point depends on both your north–south position (x) and your east–west position (y). A partial derivative answers the question: if I only move east, how fast does the elevation change? Your north–south position is locked — only x moves.

The notation for a partial derivative is ∂f/∂x — read as “del f del x” or “the partial of f with respect to x.” The symbol ∂ (called “del” or “curly d”) signals that other variables are held constant during differentiation.

The core rule: when taking ∂f/∂x, treat every variable except x as if it were a number. Then apply standard differentiation rules — power rule, product rule, chain rule — just as you would in single-variable calculus.

Worked Example: Finding ∂f/∂x and ∂f/∂y

Let’s find both partial derivatives of: f(x, y) = x³ + 3xy + y²

Step 1 — Find ∂f/∂x (treat y as a constant):

• Derivative of x³ → 3x²
• Derivative of 3xy → 3y (y is a constant, so it stays; differentiate x → 1)
• Derivative of y² → 0 (no x in this term; constant derivatives to zero)

Result: ∂f/∂x = 3x² + 3y

Step 2 — Find ∂f/∂y (treat x as a constant):

• Derivative of x³ → 0 (no y present)
• Derivative of 3xy → 3x (x is a constant; differentiate y → 1)
• Derivative of y² → 2y

Result: ∂f/∂y = 3x + 2y

Run this through the calculator above and verify — it’s a good way to build intuition before tackling harder functions.

Differentiation Rules That Apply to Partial Derivatives

The same rules you use in single-variable calculus apply here. The only difference is that non-target variables are treated as constants throughout.

Power Rule — For a term like x^n, the derivative with respect to x is n·x^(n-1). If you’re differentiating with respect to y, the entire x^n term becomes zero (it’s a constant).

Product Rule — When two variables are multiplied together (like xy), differentiate normally: ∂(xy)/∂x = y and ∂(xy)/∂y = x.

Chain Rule — For composite functions like sin(x²y), treat y as a constant when differentiating with respect to x. Differentiate the outer function first, then multiply by the derivative of the inner expression with respect to x.

Constant Rule — Any term with no instance of the target variable differentiates to zero. This is the most important rule to remember — it’s where most manual errors happen.

Second-Order and Mixed Partial Derivatives

A second-order partial derivative means you differentiate twice with respect to the same variable. For f(x, y), the second-order partial with respect to x is written ∂²f/∂x² — first find ∂f/∂x, then differentiate that result with respect to x again.

A mixed partial derivative means you differentiate with respect to two different variables in sequence. The notation ∂²f/∂y∂x means: first differentiate with respect to x, then differentiate that result with respect to y.

Here’s a useful fact worth knowing: for most well-behaved functions, the order of mixed differentiation doesn’t matter. This is called Clairaut’s Theorem — ∂²f/∂x∂y = ∂²f/∂y∂x. The mixed partial is the same regardless of which variable you differentiate first.

Use the calculator above to find second-order and mixed partials by changing the order setting before hitting Calculate.

Real-World Uses of Partial Derivatives

Partial differentiation isn’t just a classroom exercise — it shows up in fields where multiple variables interact.

Machine Learning — Every modern neural network uses partial derivatives during training. The algorithm computes the partial derivative of the loss function with respect to each weight, then updates the weights to reduce error. This process is called gradient descent.

Economics — Businesses use partial derivatives for marginal analysis. If profit depends on both labor (x) and capital (y), the partial derivative ∂P/∂x tells you exactly how much additional profit one extra unit of labor produces, holding capital fixed.

Engineering & Physics — Heat distribution, fluid flow, and wave equations all involve partial differential equations (PDEs). Partial derivatives describe how temperature or pressure changes through a material when multiple spatial dimensions are involved.

Optimization — To find the maximum or minimum of a multivariable function, you set all first-order partial derivatives equal to zero and solve the system. This is the multivariable equivalent of setting f'(x) = 0 in single-variable calculus.

Frequently Asked Questions

What is the difference between a partial derivative and a regular derivative?

A regular (ordinary) derivative applies to functions with one variable — it measures the total rate of change of f(x) as x changes. A partial derivative applies to functions with two or more variables. It measures the rate of change with respect to one variable while holding all others constant. The notation changes too: ordinary derivatives use d (as in df/dx), while partial derivatives use ∂ (as in ∂f/∂x).

How do you read the ∂ symbol?

The symbol ∂ is called “del” or “curly d.” The expression ∂f/∂x is read as “del f del x” or “the partial derivative of f with respect to x.” You’ll also see it written in subscript notation as f_x, which means the same thing.

What does “holding a variable constant” mean in practice?

It means you treat every other variable as if it were a fixed number during differentiation. If you’re finding ∂f/∂x for the function f(x, y) = 3x²y, you treat y as a constant multiplier. The derivative becomes 6xy — where y stays as part of the result because it was held constant throughout.

Can the calculator handle functions with three or more variables?

Yes. Partial differentiation works for any number of variables. For a function f(x, y, z), you can find ∂f/∂x, ∂f/∂y, or ∂f/∂z — each time treating the other two variables as constants. Enter the function and select your target variable in the calculator above.

What is a gradient vector?

The gradient of a function is the vector that contains all of its first-order partial derivatives. For f(x, y), the gradient ∇f = (∂f/∂x, ∂f/∂y). It points in the direction of the steepest uphill slope of the function at any given point. Gradient vectors are foundational in optimization, machine learning, and vector calculus.

Why is my partial derivative zero for a whole term?

Because that term contains none of your target variable. If you’re differentiating f(x, y) = x² + 5y³ with respect to x, the term 5y³ differentiates to zero — it’s a constant from x’s perspective. This is the constant rule in action and one of the most common sources of confusion for students first learning partial differentiation.

Keep Exploring Calculus and Math Tools

Partial differentiation is one piece of a larger calculus toolkit. Here are other ToolCalcPro calculators that complement it:

• Working with data and need to measure relationships between variables? The Correlation Coefficient Calculator finds Pearson, Spearman, and Kendall coefficients from raw data in seconds.
• Studying probability alongside calculus? The Binomial Distribution Calculator handles exact probability calculations without manual binomial expansion.
• Planning for the future while you study? Run your retirement numbers through the Coast FIRE Calculator — a useful break from multivariable math.

Partial derivatives power everything from AI models to economic forecasting. Use the calculator above to build your intuition one function at a time — and drop any questions in the comments below.

Have a function you’re stuck on? Share it in the comments and we’ll walk through it.