Multivariable Derivative Calculator

This calculator helps you understand multivariable derivatives by computing the function value, numerical partial derivatives, and exact partial derivatives for a specific function at a given point. The function used is f(x, y) = x³ + 2xy² - 5y + 10.

X-coordinate (x):

Y-coordinate (y):

Step Size (h) for Numerical Derivative:

Results:

Function: f(x, y) = x³ + 2xy² - 5y + 10

Exact Partial Derivative with respect to x: ∂f/∂x = 3x² + 2y²

Exact Partial Derivative with respect to y: ∂f/∂y = 4xy - 5

At Point (x, y): (, )

Function Value f(x, y):

Numerical ∂f/∂x:

Exact ∂f/∂x:

Numerical ∂f/∂y:

Exact ∂f/∂y:

Understanding Multivariable Derivatives

In calculus, a derivative measures how a function changes as its input changes. For functions with a single variable, like f(x) = x², the derivative f'(x) = 2x tells us the slope of the tangent line at any point x.

What are Multivariable Functions?

Multivariable functions are functions that depend on two or more independent variables. For example, f(x, y) = x² + y² is a function of two variables, x and y. These functions often describe surfaces in 3D space or more complex phenomena in higher dimensions.

Partial Derivatives: The Core of Multivariable Differentiation

When dealing with multivariable functions, we can't simply talk about "the" derivative because the function can change differently depending on which input variable we vary. This is where partial derivatives come in. A partial derivative measures the rate of change of a multivariable function with respect to one variable, while holding all other variables constant.

The partial derivative of f(x, y) with respect to x, denoted as ∂f/∂x, treats y as a constant and differentiates f with respect to x.
Similarly, the partial derivative of f(x, y) with respect to y, denoted as ∂f/∂y, treats x as a constant and differentiates f with respect to y.

For our example function, f(x, y) = x³ + 2xy² - 5y + 10:

To find ∂f/∂x, we treat y as a constant:
- Derivative of x³ with respect to x is 3x².
- Derivative of 2xy² with respect to x (treating 2y² as a constant) is 2y².
- Derivative of -5y with respect to x (-5y is a constant) is 0.
- Derivative of 10 with respect to x (10 is a constant) is 0.
So, ∂f/∂x = 3x² + 2y².
To find ∂f/∂y, we treat x as a constant:
- Derivative of x³ with respect to y (x³ is a constant) is 0.
- Derivative of 2xy² with respect to y (treating 2x as a constant) is 2x * (2y) = 4xy.
- Derivative of -5y with respect to y is -5.
- Derivative of 10 with respect to y (10 is a constant) is 0.
So, ∂f/∂y = 4xy - 5.

Numerical vs. Exact Derivatives

While exact partial derivatives provide the precise rate of change, sometimes it's useful or necessary to approximate them numerically. This is especially true for very complex functions or when only discrete data points are available.

The numerical partial derivative is calculated using a finite difference approximation, similar to the definition of a derivative:

Numerical ∂f/∂x ≈ (f(x + h, y) - f(x, y)) / h
Numerical ∂f/∂y ≈ (f(x, y + h) - f(x, y)) / h

Here, h is a small step size. As h approaches zero, the numerical approximation gets closer to the exact derivative. Our calculator uses a small h (e.g., 0.001) to demonstrate this approximation.

The Gradient Vector

For a multivariable function, the collection of all its first-order partial derivatives forms a vector called the gradient vector, denoted by ∇f. For a function f(x, y), the gradient is ∇f = (∂f/∂x, ∂f/∂y). The gradient vector points in the direction of the steepest ascent of the function at a given point, and its magnitude represents the rate of that steepest ascent.

Using the Calculator

Enter the desired X and Y coordinates for the point at which you want to evaluate the derivatives. You can also adjust the 'Step Size (h)' for the numerical approximation. A smaller h generally yields a more accurate numerical result, but too small an h can lead to floating-point precision issues. The calculator will then display:

The value of the function f(x, y) at your specified point.
The numerical approximation of ∂f/∂x and ∂f/∂y.
The exact values of ∂f/∂x and ∂f/∂y, calculated using the derived formulas.

Observe how the numerical results closely match the exact results when a small step size is used, illustrating the power of calculus and numerical methods.