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Partial Derivative Calculator

Calculate partial derivatives of multivariable functions.

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Partial Derivative Calculator

For multivariable functions

Partial Differentiation

A partial derivative of a multivariable function is its derivative with respect to one of those variables, with the others held constant. It measures the rate of change of the function along a particular axis.

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.

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