Free Online Calculators

Partial Derivative Calculator

Calculate partial derivatives of multivariable functions.

How the Partial Derivative Calculator Works

The Partial Derivative Calculator computes partial derivatives of multivariable functions. When a function depends on multiple variables, partial derivatives measure how the function changes with respect to one variable while holding all other variables constant. This is essential in multivariable calculus, optimization, and many areas of physics and engineering.

Understanding Partial Derivatives

For a function f(x, y), the partial derivative with respect to x, denoted ∂f/∂x or fx, measures the rate of change in the x-direction while y remains fixed. Similarly, ∂f/∂y measures change in the y-direction. Unlike ordinary derivatives, we must specify which variable we're differentiating with respect to.

Notation

  • ∂f/∂x - Leibniz notation for partial derivative with respect to x
  • fx - subscript notation
  • fx(x,y) - partial derivative as a function
  • ∂2f/∂x2 - second partial derivative with respect to x
  • ∂2f/∂x∂y - mixed partial derivative (differentiate first by y, then by x)

How to Compute Partial Derivatives

  1. Identify which variable you're differentiating with respect to
  2. Treat all other variables as constants
  3. Apply ordinary differentiation rules to your chosen variable
  4. Simplify the result

Important Properties

  • Clairaut's Theorem: For continuous second partial derivatives, ∂2f/∂x∂y = ∂2f/∂y∂x (mixed partials are equal)
  • Linearity: ∂/∂x[af + bg] = a(∂f/∂x) + b(∂g/∂x)
  • Product Rule: ∂/∂x[f·g] = (∂f/∂x)·g + f·(∂g/∂x)
  • Chain Rule: For composite functions, partial derivatives combine according to the multivariable chain rule

Examples

Partial Derivative Examples

Example 1: Basic Polynomial

Problem: Find ∂f/∂x and ∂f/∂y for f(x,y) = x2y + 3xy2 - 5y

Solution:

  • ∂f/∂x (treat y as constant): d/dx[x2y] + d/dx[3xy2] + d/dx[-5y]
  • ∂f/∂x = 2xy + 3y2 + 0 = 2xy + 3y2
  • ∂f/∂y (treat x as constant): d/dy[x2y] + d/dy[3xy2] + d/dy[-5y]
  • ∂f/∂y = x2 + 6xy - 5
Example 2: Exponential Function

Problem: Find ∂f/∂x for f(x,y) = e^(xy)

Solution:

  • Use chain rule: e^(xy) · ∂/∂x[xy]
  • ∂/∂x[xy] = y (treating y as constant)
  • Result: ∂f/∂x = y·e^(xy)
Example 3: Trigonometric Function

Problem: Find ∂f/∂x and ∂f/∂y for f(x,y) = sin(x)cos(y)

Solution:

  • ∂f/∂x: differentiate sin(x), keep cos(y) as constant factor
  • ∂f/∂x = cos(x)cos(y)
  • ∂f/∂y: keep sin(x) as constant, differentiate cos(y)
  • ∂f/∂y = sin(x)(-sin(y)) = -sin(x)sin(y)
Example 4: Second Partial Derivatives

Problem: Find all second partials of f(x,y) = x3y2

Solution:

  • First partials: ∂f/∂x = 3x2y2, ∂f/∂y = 2x3y
  • ∂2f/∂x2 = ∂/∂x[3x2y2] = 6xy2
  • ∂2f/∂y2 = ∂/∂y[2x3y] = 2x3
  • ∂2f/∂x∂y = ∂/∂x[2x3y] = 6x2y
  • ∂2f/∂y∂x = ∂/∂y[3x2y2] = 6x2y (equal to ∂2f/∂x∂y, as expected)
Example 5: Three Variables

Problem: Find ∂f/∂z for f(x,y,z) = x2y + yz2 + xz

Solution:

  • Treat x and y as constants, differentiate with respect to z
  • ∂/∂z[x2y] = 0 (no z terms)
  • ∂/∂z[yz2] = 2yz (y is constant)
  • ∂/∂z[xz] = x (x is constant)
  • Result: ∂f/∂z = 2yz + x
Example 6: Implicit Differentiation

Problem: If x2+ y2 + z2 = 25, find ∂z/∂x

Solution:

  • Take partial derivative of both sides with respect to x (y is constant)
  • 2x + 0 + 2z(∂z/∂x) = 0
  • Solve for ∂z/∂x: 2z(∂z/∂x) = -2x
  • Result: ∂z/∂x = -x/z

Tips & Best Practices

Tips for Computing Partial Derivatives

  • Treat Other Variables as Constants: This is the key concept. When finding ∂f/∂x, pretend y, z, etc. are just numbers.
  • Be Clear About Notation: Always specify which variable you're differentiating with respect to using proper notation.
  • Check Mixed Partials: Verify Clairaut's theorem: ∂2f/∂x∂y should equal ∂2f/∂y∂x for continuous functions.
  • Use Gradient Notation: The gradient ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) collects all partial derivatives into a vector.
  • Practice Chain Rule: The multivariable chain rule is more complex than the single-variable version; practice is essential.
  • Understand Geometric Meaning: ∂f/∂x is the slope of the surface f(x,y) in the x-direction at a point.
  • Order Matters in Notation: In ∂2f/∂x∂y, read from right to left: differentiate by y first, then by x.
  • Critical Points: Points where all first partials equal zero are critical points, candidates for maxima, minima, or saddle points.
  • Directional Derivatives: To find rate of change in any direction, use ∇f · û where û is a unit direction vector.
  • Verify Your Work: Check dimensions and limiting cases to ensure your answer makes physical sense.

Frequently Asked Questions

How to use the Partial Derivative Calculator

Follow these steps to get accurate results with the partial derivative calculator.

  1. 1

    Enter your values

    Fill in the required input fields above. Units can be changed where available.

  2. 2

    Click Calculate

    Press the calculate button to compute results instantly in your browser.

  3. 3

    Review your results

    View the computed outputs and use related calculators for deeper analysis.

Related Calculators

Explore these related calculators to help you make informed decisions:

Derivative Calculator

Calculate derivatives of functions with step-by-step solutions.

Calculus Calculator

Perform calculus operations including derivatives, integrals, and limits.

Popular Calculators

BMI Calculator

Health

Calculate your Body Mass Index

Mortgage Calculator

Financial

Estimate monthly mortgage payments

Percentage Calculator

Math

Calculate percentages easily

Loan Calculator

Financial

Calculate loan payments and interest

Calorie Calculator

Health

Calculate daily calorie needs

Compound Interest Calculator

Financial

Calculate compound interest growth

View All Calculators