Question

Find the first partial derivatives of at the point .

Answer 1

The first partial derivatives of f(x, y) at the point (a, b) are
`\( (\partial f)/(\partial x)(a, b) \)` and
`\( (\partial f)/(\partial y)(a, b) \)`.

Step-by-step explanation:

For a function f(x, y), the partial derivatives with respect to x and y, denoted as
`\( (\partial f)/(\partial x) \) and \( (\partial f)/(\partial y) \)`, respectively, indicate how the function changes concerning each variable while holding the other constant. To find these derivatives at a specific point (a, b), one computes
`\( (\partial f)/(\partial x)(a, b) \)` and
`\( (\partial f)/(\partial y)(a, b) \)`. The former represents the rate of change of f concerning x at (a, b), while the latter signifies the rate of change concerning y at the same point.

These partial derivatives can be calculated using techniques such as the limit definition of derivatives or specific rules depending on the function provided. For instance, if f(x, y) is a function like f(x, y) = x² + 3xy - 2y, then
`\( (\partial f)/(\partial x) \)` would be 2x + 3y and
`\( (\partial f)/(\partial y) \)` would be
`\( 3x - 2 \)`.

Evaluating these expressions at the point (a, b) involves substituting a for x and b for y in each respective derivative expression.

Here is complete question;

"Find the first partial derivatives of f(x, y) at the point (a, b), where f(x, y) is [insert the function], and (a, b) is [insert the point]."