To find the first partial derivatives of the function f(x, y), you need to follow these general steps:
- Identify the Function: Start by clearly defining the function f(x, y) for which you want to calculate the partial derivatives.
- Determine the Variables: Recognize that x and y are the independent variables in the function.
- Calculate the Partial Derivative with Respect to x: To find the first partial derivative of f with respect to x, denoted as ∂f/∂x, you will treat y as a constant. This means you differentiate f(x, y) as if y is a fixed value:
∂f/∂x = lim (h → 0) [f(x + h, y) - f(x, y)] / h - Calculate the Partial Derivative with Respect to y: Similarly, to find the first partial derivative of f with respect to y, denoted as ∂f/∂y, you will treat x as a constant. This involves differentiating f(x, y) assuming x remains unchanged:
∂f/∂y = lim (k → 0) [f(x, y + k) - f(x, y)] / k
With these derivatives calculated, you will obtain two expressions: ∂f/∂x and ∂f/∂y. These represent the rates of change of f with respect to each variable, providing insights into how the function behaves as each variable changes independently.
Example: Let’s illustrate this with a specific function: consider f(x, y) = x²y + 3xy². To find the partial derivatives:
- For ∂f/∂x:
Differentiating while treating y as constant gives:∂f/∂x = 2xy + 3y² - For ∂f/∂y:
Differentiating while treating x as constant results in:∂f/∂y = x² + 6xy
In conclusion, understanding and calculating partial derivatives is essential in multivariable calculus, as it helps in analyzing functions that depend on multiple variables.