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Partial derivative calculator

Find a partial derivative

Choose one variable to change while every other independent variable stays fixed, then inspect the exact derivative and its conditions.

Use the math keyboard or your device keyboard.
Try an example

Result

Partial derivative with respect to x

∂∂x(x2y+y3)=2xy\frac{\partial}{\partial x}(x^2y+y^3)=2xy
Conditions
  • x and y are independent real variables.
  • y is held constant while differentiating with respect to x.

Steps

  1. Fix the other variables. Treat y as a constant for this x-partial derivative.∂y∂x=0\frac{\partial y}{\partial x}=0
  2. Differentiate the x-term. Apply the power rule to x squared and keep the constant factor y.∂∂x(x2y)=2xy\frac{\partial}{\partial x}(x^2y)=2xy
  3. Differentiate the constant term. The term y cubed does not change when only x changes.∂∂x(y3)=0\frac{\partial}{\partial x}(y^3)=0
Independent check

Holding y fixed, the centered difference quotient in the x direction approaches 2xy at regular sample points.

Scope

What this partial derivative calculator covers

A partial derivative measures how a multivariable function changes in one coordinate direction. This calculator differentiates with respect to the selected variable and treats the other independent variables as constants.

Choose x, y, or z

Differentiate with respect to one selected coordinate while treating the other independent variables as constants.

Examples: x-partial derivatives, y-partial derivatives, z-partial derivatives

Algebraic combinations

Apply sum, product, quotient, and power rules to supported multivariable polynomials and rational expressions.

Examples: x squared times y, a quotient containing x and y

Composite functions

Use the chain rule for supported exponential, logarithmic, and trigonometric expressions whose inner function contains several variables.

Examples: sine of xy, logarithm of x squared plus y squared

Point evaluation after differentiation

Find the symbolic partial derivative first, then substitute coordinates only where the original function and derivative are defined.

Examples: surface slope at a point, coordinate-direction rate

How to use it

Enter enough information for one clear task

  1. 1
    Enter the function

    Use explicit grouping for products, exponents, logarithms, and denominators. Enter only the function, not a sentence describing the task.

  2. 2
    Choose the differentiation variable

    The selected symbol changes. Every other independent variable is held constant during this derivative.

  3. 3
    Confirm the interpretation

    Check the displayed variable and make sure the input is a multivariable function rather than an equation defining one variable implicitly.

  4. 4
    Check the domain

    Carry restrictions from logarithms, roots, and denominators into the result, then compare a directional difference quotient at a valid point.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

Differentiate with respect to x

hold y constant

∂∂x(x2y+3y2)\frac{\partial}{\partial x}(x^2y+3y^2)

Expected result

2xy2xy

Differentiate with respect to y

hold x constant

∂∂y(x2y+3y2)\frac{\partial}{\partial y}(x^2y+3y^2)

Expected result

x2+6yx^2+6y

Exponential in three variables

chain and sum rules

∂∂z(exz+ysin⁡z)\frac{\partial}{\partial z}\left(e^{xz}+y\sin z\right)

Expected result

xexz+ycos⁡zxe^{xz}+y\cos z

Logarithmic surface

chain rule

∂∂xln⁡(x2+y2)\frac{\partial}{\partial x}\ln(x^2+y^2)

Expected result

2xx2+y2\frac{2x}{x^2+y^2}

Trigonometric composition

chain rule with x fixed

∂∂ysin⁡(xy)\frac{\partial}{\partial y}\sin(xy)

Expected result

xcos⁡(xy)x\cos(xy)

Rational expression

quotient rule with y fixed

∂∂x(x+yx−y)\frac{\partial}{\partial x}\left(\frac{x+y}{x-y}\right)

Expected result

−2y(x−y)2\frac{-2y}{(x-y)^2}

Complete example

Differentiate a polynomial-exponential surface

Only x changes. The symbol y acts as a constant coefficient in the polynomial and as a constant chain-rule factor in the exponential.

f(x,y)=x2y+exy,∂f∂xf(x,y)=x^2y+e^{xy},\qquad \frac{\partial f}{\partial x}
  1. 1
    Differentiate the polynomial term

    Keep y and apply the power rule to x squared.

    ∂∂x(x2y)=2xy\frac{\partial}{\partial x}(x^2y)=2xy
  2. 2
    Differentiate the exponential term

    The derivative of the exponent xy with respect to x is y.

    ∂∂xexy=yexy\frac{\partial}{\partial x}e^{xy}=ye^{xy}
  3. 3
    Combine the terms

    Linearity lets the two partial derivatives be added.

    fx(x,y)=2xy+yexyf_x(x,y)=2xy+ye^{xy}
  4. 4
    Check a point

    At x = 1 and y = 2, the formula gives 4 plus 2e squared.

    fx(1,2)=4+2e2f_x(1,2)=4+2e^2
∂f∂x=2xy+yexy\frac{\partial f}{\partial x}=2xy+ye^{xy}

Verification: With y fixed at 2, a centered difference quotient of f(x,2) at x = 1 approaches 4 + 2e squared, matching the symbolic result.

Avoidable errors

Common mistakes and how to fix them

Differentiating every variable

Problem: Applying a derivative to y while finding the x-partial derivative.

Why it matters: Independent variables other than the selected variable are fixed for a partial derivative.

Better approach: State the active variable first and mark every other independent variable as constant.

Freezing the selected variable

Problem: Treating x squared as constant while computing a partial derivative with respect to x.

Why it matters: The selected coordinate is the one direction in which the function is allowed to change.

Better approach: Highlight each occurrence of the selected variable before applying derivative rules.

Substituting coordinates too early

Problem: Replacing x and y with numbers before forming the symbolic derivative.

Why it matters: Early substitution can erase the local change that the derivative is meant to measure.

Better approach: Differentiate symbolically, preserve the domain, and evaluate the point last.

Using implicit differentiation by accident

Problem: Adding dy over dx factors to a function where x and y are independent inputs.

Why it matters: Implicit differentiation treats y as dependent on x, which is a different mathematical task.

Better approach: Use the implicit differentiation calculator for an equation that relates x and y.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • Hold every nonselected variable fixed and compare a centered difference quotient in the selected coordinate direction.
  • Differentiate the simplified and unsimplified forms independently when algebraic cancellation is involved.
  • Evaluate only at points inside the original domain, including logarithm, radical, and denominator restrictions.

When to stop and revise the input

  • Use the implicit differentiation calculator when the input is an equation that defines one variable through another.
  • Do not infer total differentiability from the existence of one or even all partial derivatives at a point.
  • Revise input whose variable roles, grouping, or real-versus-complex domain are ambiguous.

Common questions

Find a partial derivative FAQ

What does a partial derivative hold constant?

It holds every independent variable except the selected differentiation variable constant. For f of x and y, the x-partial changes x while y stays fixed.

Is a partial derivative the same as implicit differentiation?

No. A partial derivative treats x and y as independent inputs to a multivariable function. Implicit differentiation treats y as a function of x because an equation relates them.

Can I evaluate a partial derivative at a point?

Yes. Differentiate first, then substitute the coordinates, provided the original function and the partial derivative are defined at that point.

Does having partial derivatives mean a function is differentiable?

Not always. Partial derivatives can exist at a point even when the function is not differentiable there. Continuity of the partial derivatives near the point is a common sufficient condition, but not a necessary one.

Sources and curriculum alignment

This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.