Partial Derivative Evaluator
This calculator evaluates the partial derivatives of the function f(x, y) = x³ + 2xy² + 5y at a specified point (x, y).
Results:
For f(x, y) = x³ + 2xy² + 5y:
Partial derivative with respect to x (∂f/∂x) = 3x² + 2y²
Evaluated ∂f/∂x at (x, y):
Partial derivative with respect to y (∂f/∂y) = 4xy + 5
Evaluated ∂f/∂y at (x, y):
Understanding Partial Differentiation
Partial differentiation is a fundamental concept in multivariable calculus that allows us to study the rate of change of a function with respect to one variable, while holding all other variables constant. This is crucial when dealing with functions that depend on two or more independent variables, such as temperature distribution across a surface, pressure in a gas, or economic models.
What is a Partial Derivative?
Imagine a function f(x, y) that describes a surface in 3D space. If you want to know how steeply the surface is rising or falling as you move only in the x-direction (parallel to the x-axis), you would use a partial derivative with respect to x, denoted as ∂f/∂x. When calculating ∂f/∂x, you treat y as a constant. Similarly, if you move only in the y-direction, you would use ∂f/∂y, treating x as a constant.
How to Calculate Partial Derivatives (Conceptually)
The process is similar to ordinary differentiation. You apply all the standard differentiation rules (power rule, product rule, chain rule, etc.) but only to the variable you are differentiating with respect to. Any other variables are treated as constants.
Example Function and its Partial Derivatives:
Let's consider the function:
f(x, y) = x³ + 2xy² + 5y
To find the partial derivative with respect to x (∂f/∂x), we treat y as a constant:
- Derivative of
x³with respect to x is3x². - Derivative of
2xy²with respect to x (treating2y²as a constant coefficient) is2y² * (derivative of x with respect to x) = 2y² * 1 = 2y². - Derivative of
5ywith respect to x (treating5yas a constant) is0.
So, ∂f/∂x = 3x² + 2y²
To find the partial derivative with respect to y (∂f/∂y), we treat x as a constant:
- Derivative of
x³with respect to y (treatingx³as a constant) is0. - Derivative of
2xy²with respect to y (treating2xas a constant coefficient) is2x * (derivative of y² with respect to y) = 2x * 2y = 4xy. - Derivative of
5ywith respect to y is5.
So, ∂f/∂y = 4xy + 5
Using the Calculator
Our calculator allows you to input specific values for x and y and then evaluates these pre-calculated partial derivatives at that exact point. This helps you understand the instantaneous rate of change of the function in the x and y directions at any given point.
Example Calculation:
Let's use the calculator with x = 2 and y = 3.
- For ∂f/∂x:
∂f/∂x = 3x² + 2y²
∂f/∂x = 3(2)² + 2(3)²
∂f/∂x = 3(4) + 2(9)
∂f/∂x = 12 + 18 = 30 - For ∂f/∂y:
∂f/∂y = 4xy + 5
∂f/∂y = 4(2)(3) + 5
∂f/∂y = 24 + 5 = 29
The calculator will output these values, showing the rate of change of f(x, y) at the point (2, 3) in the x and y directions, respectively.