Partial Derivative Calculator
Calculate ∂f/∂x or ∂f/∂y for multivariable functions with step-by-step reasoning, sample values, and a diagram that shows the slope visually.
Interpreting this as: f(x,y) = x^2 · y + 3 · x · y^2
Use x and y explicitly. For powers type x² or y³. Enter pi, e, sqrt, sin, cos, tan, ln, log, exp, and abs. The calculator computes either ∂f/∂x or ∂f/∂y and shows the formula with a matching diagram.
Use these examples
This is the rate of change of f in the x-direction at the selected point.
Visual partial derivative diagram
The top chart shows the function slice along the chosen variable. The tangent line at the selected point is labeled with the actual slope value from the partial derivative.
Partial derivative steps
Each rule is shown explicitly so you can follow the exact reasoning behind the result.
Given: Compute ∂f/∂x for f(x,y) = x²*y + 3*x*y² at (1, 2)
Step 1: Power rule for ∂/∂x
Apply the power rule. If the base itself depends on x, multiply by its derivative too.
∂/∂x[x²] = 2·x¹Step 2: Constant multiple rule for ∂/∂x
y is independent of x, so it remains constant while differentiating the first factor.
∂/∂x[x² · y] = (2 · x) · yStep 3: Constant multiple rule for ∂/∂x
3 is independent of x, so it remains constant while differentiating the other factor.
∂/∂x[3 · x] = 3 · (1)Step 4: Constant multiple rule for ∂/∂x
y^2 is independent of x, so it remains constant while differentiating the first factor.
∂/∂x[3 · x · y²] = (3) · y²Step 5: Sum rule for ∂/∂x
The partial derivative of a sum is the sum of the partial derivatives of each term.
∂/∂x[x² · y + 3 · x · y²] = 2 · x · y + 3 · y²Step 6: Simplify the partial derivative result
Write the final partial derivative in its simplest form.
∂f/∂x = 2 · x · y + 3 · y²
Final partial derivative
∂f/∂x = 2 · x · y + 3 · y²
Partial Derivative Calculator with step-by-step solutions
Use this Partial Derivative Calculator to find ∂f/∂x or ∂f/∂y for any function of x and y. Enter a multivariable formula such as x^2*y + 3*x*y^2, choose the variable to differentiate, and the calculator returns the symbolic partial derivative with a full explanation of each rule used.
The design follows the same step-by-step approach as the Limit Calculator, but for multivariable calculus. Instead of sending x alone toward a point, this calculator holds one variable fixed and differentiates the other variable as a partial derivative, then shows the result as a formula and as a numerical slope at a chosen coordinate.
The content on this page also includes practical formulas, calculator keywords, and a clear graphic so the answer is easy to read from the diagram itself. Every label on the diagram is chosen to make the meaning of ∂f/∂x and ∂f/∂y obvious even if you are learning partial derivatives for the first time.
Why partial derivatives matter
A partial derivative measures how a function changes when only one variable moves and the others remain fixed. For example, ∂f/∂x describes the slope of f when y is held constant, while ∂f/∂y describes the slope when x stays fixed. That means partial derivatives are the most important tool in multivariable calculus, optimization, economics, physics, and machine learning.
This calculator supports the most common partial derivative formulas: product rule, quotient rule, power rule, chain rule, and derivatives of sine, cosine, exponential, logarithmic, and root functions. The step-by-step solution shows which rule is used at every stage, just like the Limit Calculator shows whether it used direct substitution or L'Hôpital's Rule.
For a function of two variables, the diagram below fixes one variable and displays the slice along the other variable. The tangent line and point labels show both the function value and the partial derivative value, so you can see directly how the rate of change is computed.
How to use this partial derivative formula calculator
Type your function in terms of x and y, such as x^2*y + 3*x*y^2 or sin(x*y) + ln(x^2 + y^2). Use x and y for the variables and power notation with ^, for example x^3 for x cubed. The calculator understands standard functions like sin(x), cos(x), tan(x), ln(x), log(x), sqrt(x), exp(x), and abs(x).
Choose the variable to differentiate: select x to compute ∂f/∂x, or select y to compute ∂f/∂y. The calculator will hold the other variable constant automatically and apply the correct partial derivative rules.
If you want a numerical answer at a point, enter a coordinate such as x = 1 and y = 2. The calculator then shows both the symbolic derivative formula and the numeric slope at that coordinate, just like a partial derivative example in a textbook.
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From formula to diagram: understanding the slice and slope
This calculator makes partial derivatives visual by drawing the function slice that corresponds to the selected variable. If you choose ∂/∂x, the diagram shows f(x, y0) for a fixed y value. If you choose ∂/∂y, it shows f(x0, y) for a fixed x value.
The diagram labels the point where the partial derivative is evaluated and the slope of the tangent line at that point. That means the diagram itself carries the values, not just the text. Each diagram label is designed to explain the formula, the coordinate, and the result without confusion.
For example, if f(x,y) = x^2 y + 3 x y^2 and you evaluate ∂f/∂x at (1,2), the calculator shows the slice for y = 2, the function value at x = 1, and the slope value ∂f/∂x = 13 at that point. You can watch that rate of change as a tangent line drawn directly on the slice.