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Partial Derivative Calculator

Compute partial derivatives of multivariable functions. Treats all other variables as constants with step-by-step solutions.

Back to Derivative Calculator

Enter a math expression using standard notation. Use * for multiplication, ^ for exponents.

Variable of differentiation. For partial derivatives, change to y, z, etc.

Derivative Calculator Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Power Rule

Differentiate a polynomial using the power rule.

Key values: f(x) = x^3 - 2x^2 + x · 1st derivative

Chain Rule

Differentiate a composite function with sin(x^2).

Key values: f(x) = sin(x^2) · Chain rule applied

Product Rule with Tangent

Differentiate exp(x)*cos(x) and evaluate the tangent at x = 0.

Key values: f(x) = exp(x)*cos(x) · Tangent at x = 0

Second Derivative

Compute the second derivative to analyze concavity.

Key values: f(x) = x^4 - 6x^2 + 4 · 2nd derivative

Documentation

What Are Partial Derivatives?

When a function depends on multiple variables, a partial derivative measures how the function changes with respect to one variable while holding all others constant. The notation ∂f/∂x\partial f / \partial x uses the "curly d" to distinguish it from ordinary derivatives.

For f(x,y)=x2y+y3f(x, y) = x^2y + y^3:

∂f∂x=2xy∂f∂y=x2+3y2\frac{\partial f}{\partial x} = 2xy \qquad \frac{\partial f}{\partial y} = x^2 + 3y^2

To compute ∂f/∂x\partial f/\partial x, treat yy as a constant and differentiate normally with respect to xx.

The Gradient Vector

Collecting all partial derivatives into a vector gives the gradient:

∇f=(∂f∂x,∂f∂y,…)\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \ldots\right)

The gradient points in the direction of steepest ascent. Its magnitude gives the rate of change in that direction. This is the foundation of gradient descent in machine learning.

Higher-Order and Mixed Partials

Second-order partial derivatives include both "pure" and "mixed" partials:

∂2f∂x2,∂2f∂y2,∂2f∂x∂y\frac{\partial^2 f}{\partial x^2}, \quad \frac{\partial^2 f}{\partial y^2}, \quad \frac{\partial^2 f}{\partial x \partial y}

Clairaut's theorem: If the mixed partials are continuous, then the order of differentiation doesn't matter: ∂2f∂x∂y=∂2f∂y∂x\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}.

Applications

  • Optimization: Critical points occur where all partial derivatives equal zero. The second partial derivative test uses the Hessian matrix to classify them.
  • Physics: Heat equation ∂u/∂t=k∇2u\partial u/\partial t = k \nabla^2 u, wave equation, Maxwell's equations — all expressed via partial derivatives.
  • Economics: Marginal cost, marginal utility, and elasticity are all partial derivatives of multivariable economic functions.

To compute partial derivatives with this calculator, change the variable field to the variable you want to differentiate with respect to. All other symbols are treated as constants automatically.

Frequently Asked Questions

What is a partial derivative?

A partial derivative measures how a multivariable function changes with respect to one variable while holding all other variables constant. The notation uses the curly d symbol: ∂f/∂x\partial f / \partial x means differentiate with respect to xx, treating yy, zz, etc. as constants.

How do I compute a partial derivative?

Treat every variable except the one you are differentiating with respect to as a constant, then differentiate normally. For f(x,y)=x2y+y3f(x, y) = x^2 y + y^3, the partial with respect to xx is 2xy2xy (yy treated as constant), and with respect to yy it is x2+3y2x^2 + 3y^2.

What is the gradient vector and why does it matter?

The gradient is the vector of all partial derivatives: ∇f=(∂f/∂x,∂f/∂y,…)\nabla f = (\partial f/\partial x, \partial f/\partial y, \ldots). It points in the direction of steepest ascent and its magnitude is the rate of change in that direction. Gradient descent -- the core of machine learning optimization -- moves opposite to the gradient.

What is Clairaut's theorem about mixed partial derivatives?

Clairaut's theorem states that if the mixed partial derivatives are continuous, the order of differentiation does not matter: ∂2f∂x∂y=∂2f∂y∂x\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}. This symmetry simplifies many multivariable calculus computations.

Where are partial derivatives used in practice?

Partial derivatives appear in optimization (finding critical points of multivariable functions), physics (heat equation, wave equation, Maxwell's equations), economics (marginal cost, marginal utility), and machine learning (backpropagation computes partial derivatives of the loss function).

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