Limit Definition of Partial Derivatives: Exploring f_x and f_y at -21

Understanding the limit definition of partial derivatives is crucial in multivariable calculus. This concept is not only fundamental in advanced mathematics but also essential in fields such as physics, engineering, and economics. The partial derivatives of a function (f(x, y)) with respect to (x) and (y) at a point can be determined through the limit process. In this article, we will explore how to compute the partial derivatives of (f(x, y) 42x - 3y - xy^2) at the point ((-21, -21)).

Understanding Partial Derivatives

Partial derivatives measure the rate of change of a multivariable function with respect to one of its variables, while keeping the other variables constant. The symbol for the partial derivative of a function with respect to (x) is (frac{partial f}{partial x}), and with respect to (y) is (frac{partial f}{partial y}). These derivatives are computed using the limit definition, as shown in the following section.

Limit Definition of Partial Derivatives

The limit definition of the partial derivative of a function (f(x, y)) with respect to (x) at a point ((x_0, y_0)) is:

[frac{partial f}{partial x}bigg|_{(x_0, y_0)} lim_{h to 0} frac{f(x_0 h, y_0) - f(x_0, y_0)}{h}]

Similarly, the limit definition of the partial derivative with respect to (y) at the same point is:

[frac{partial f}{partial y}bigg|_{(x_0, y_0)} lim_{k to 0} frac{f(x_0, y_0 k) - f(x_0, y_0)}{k}]

In our specific case, we need to compute (frac{partial f}{partial x}bigg|_{(-21, -21)}) and (frac{partial f}{partial y}bigg|_{(-21, -21)}) using these definitions.

Computing the Partial Derivative with Respect to x

To compute (frac{partial f}{partial x}bigg|_{(-21, -21)}), we will use the limit definition:

[frac{partial f}{partial x}bigg|_{(-21, -21)} lim_{h to 0} frac{f(-21 h, -21) - f(-21, -21)}{h}]

Substitute (f(x, y) 42x - 3y - xy^2) into the equation:

[ lim_{h to 0} frac{42(-21 h) - 3(-21) - (-21 h)(-21)^2 - (42(-21) - 3(-21) - (-21)(-21)^2)}{h}]

[ lim_{h to 0} frac{-882 42h 63 - (-21 h)441 - (-882 63)}{h}]

[ lim_{h to 0} frac{-882 42h 63 - 21 times 441 h times 441 - 882 63}{h}]

[ lim_{h to 0} frac{42h - 21 times 441}{h}]

[ 42 - 21 times 441 -9069]

Computing the Partial Derivative with Respect to y

To compute (frac{partial f}{partial y}bigg|_{(-21, -21)}), we will use the limit definition:

[frac{partial f}{partial y}bigg|_{(-21, -21)} lim_{k to 0} frac{f(-21, -21 k) - f(-21, -21)}{k}]

Substitute (f(x, y) 42x - 3y - xy^2) into the equation:

[ lim_{k to 0} frac{42(-21) - 3(-21 k) - (-21)(-21 k)^2 - (42(-21) - 3(-21) - (-21)(-21)^2)}{k}]

[ lim_{k to 0} frac{-882 3 3k - (-21)(-21 2k - k^2) - 882 63}{k}]

[ lim_{k to 0} frac{-882 3 3k 21(-21 2k - k^2) - 882 63}{k}]

[ lim_{k to 0} frac{-882 3 3k - 441 42k - 21k^2 - 882 63}{k}]

[ lim_{k to 0} frac{42k - 21k^2 - 1221}{k}]

[ 42 - 21 times 21 -441]

Conclusion

In conclusion, we have used the limit definition of partial derivatives to compute (frac{partial f}{partial x}bigg|_{(-21, -21)} -9069) and (frac{partial f}{partial y}bigg|_{(-21, -21)} -441) for the function (f(x, y) 42x - 3y - xy^2). Understanding and applying the limit definition of partial derivatives is essential for grasping the behavior of multivariable functions and has numerous practical applications in various fields.