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Calcimator

Partial Derivative Calculator

Calculate partial derivatives of f(x,y) = axy + bx² + cy² + d. Find df/dx, df/dy, and the gradient magnitude.

Inputs

Results

∂f/∂x

5

∂f/∂y

7

f(x,y)6
|∇f|8.6023
How to Use This Calculator
  1. Select the function type (e.g., f(x,y) = axⁿyᵐ or a product/composition of x and y terms).
  2. Enter the function parameters and the point (x₀, y₀) at which to evaluate the partial derivatives.
  3. ∂f/∂x is computed by treating y as a constant and differentiating with respect to x only.
  4. ∂f/∂y is computed by treating x as a constant and differentiating with respect to y only.
  5. Together, (∂f/∂x, ∂f/∂y) form the gradient vector ∇f, which the Gradient Calculator can use directly.
  6. Mixed partial ∂²f/∂x∂y is shown where applicable; by Clairaut's theorem it equals ∂²f/∂y∂x for smooth functions.

How the result changes with Coefficient of x² (b)

Coefficient of x² (b)∂f/∂x∂f/∂y
-16-317
-6-117
6137
16337

What each input means

Coefficient of xy (a)
Coefficient of the xy cross-term
Coefficient of x² (b)
Coefficient of x²
Coefficient of y² (c)
Coefficient of y²
Constant (d)
Constant term
x Value
Point to evaluate at (x coordinate)
y Value
Point to evaluate at (y coordinate)

How this is calculated

Worked example, using the default values

  1. Identify Input Parameters
    4 parameters
    Coefficient of xy (a) = 1, Coefficient of x² (b) = 2, Coefficient of y² (c) = 3, Constant (d) = 0 = 6 input(s) provided
  2. Calculate ∂f/∂x
    ∂f/∂x
    5 = 5
  3. Calculate ∂f/∂y
    ∂f/∂y
    7 = 7
  4. Calculate f
    f
    6 = 6
  5. Calculate |∇f|
    |∇f|
    8.6023 = 8.6023

Engine last updated . Checked against 3 independently-derived tests — how we verify calculators.

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