91Ó°ÊÓ

Multivariable functions are equations that involve more than one variable. In other words, these functions take multiple inputs to produce an output. For example, the function given in the exercise is defined as:
\( f(x, y) = 2x - 5xy \).
Here, the two variables are \( x \) and \( y \), which affect the output value of the function. Understanding multivariable functions is crucial in many fields of science and engineering. These functions help describe phenomena where multiple factors come into play.
When working with such functions, it's important to understand how changing one variable while keeping others constant can affect the output.

Partial differentiation involves finding the derivative of a multivariable function with respect to one variable, while keeping the other variables constant. This process helps us understand how a function changes as we vary one particular variable. For example:
1. To find the partial derivative with respect to \( x \), we treat \( y \) as a constant and differentiate the function \( f(x, y) = 2x - 5xy \). This yields:
\[ f_x = \frac{\text{d}}{\text{d}x}(2x - 5xy) = 2 - 5y \]
2. Similarly, to find the partial derivative with respect to \( y \), treat \( x \) as a constant and differentiate, yielding: \[ f_y = \frac{\text{d}}{\text{d}y}(2x - 5xy) = -5x \]
Understanding partial differentiation is essential for analyzing how changes in one independent variable impact a function, keeping other variables fixed.

Once partial derivatives are found, we can evaluate them at specific points. This means plugging in specific values for the variables to find the derivative's value at those points.
In our exercise, we need to evaluate:
1. \( f_x(-2, 4) \)
Using \( f_x = 2 - 5y \), substitute \( y = 4 \): \[ f_x(-2, 4) = 2 - 5(4) = 2 - 20 = -18 \] Thus, \( f_x(-2, 4) = -18 \)
2. \( f_y(4, -3) \)
Using \( f_y = -5x \), substitute \( x = 4 \): \[ f_y(4, -3) = -5(4) = -20 \] Thus, \( f_y(4, -3) = -20 \)
Evaluating partial derivatives at given points helps understand the function's behavior near those points. It gives concrete values showing how much the function changes as one variable changes, while the other is fixed.