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Partial derivatives are a core part of multivariable calculus and essential for understanding how multivariable functions change. When we have a function like \( f(x, y) = \sqrt{5 - x^2 - y^2} \), it depends on both \( x \) and \( y \). Each of these variables contributes to the overall behavior of the function.To find the partial derivative with respect to \( x \), denoted as \( \frac{\partial f}{\partial x} \), we treat \( y \) as a constant and differentiate only regarding \( x \). Similarly, for \( \frac{\partial f}{\partial y} \), we treat \( x \) as constant and differentiate with respect to \( y \).- **Example:** In the original solution, the partial derivatives are: - \( \frac{\partial f}{\partial x} = \frac{-x}{\sqrt{5 - x^2 - y^2}} \) - \( \frac{\partial f}{\partial y} = \frac{-y}{\sqrt{5 - x^2 - y^2}} \)These derivatives tell us the rate at which \( f \) changes as \( x \) or \( y \) changes individually. They are vital in computing the gradient vector, which shows the direction of quickest increase.

The rate of change is a concept that tells us how fast a function's value increases or decreases relative to changes in its inputs. For functions that take more than one variable, the gradient vector plays a crucial role.In the context of the original problem, the fastest rate of change of the function \( f(x, y) = \sqrt{5-x^2-y^2} \) at the point \( (1,2) \) is determined by the magnitude of its gradient vector.- **Gradient:** Think of the gradient as a special vector that points in the direction of the steepest ascent of a function.- **Rate Determination:** The magnitude of the gradient vector, \( \| abla f \| \), gives us the rate of increase in that direction.To compute the rate of increase for this function:- Calculate \( \| abla f \| \), which was found to be \( \sqrt{5} \).- This value \( \sqrt{5} \) represents how fast the function increases in the direction of the gradient vector at the point \( (1,2) \).

A unit vector is a vector that has a length of one. In the context of gradients, unit vectors are used to specify a direction without regard to magnitude.When working with gradients, the gradient vector itself points in the direction of the steepest increase, but it is not necessarily a unit vector. To determine just the direction, irrespective of the magnitude, we convert the gradient into a unit vector.- **Normalization Process:** Transform the gradient vector into a unit vector by dividing each component of the vector by its magnitude. This ensures that the resulting vector has a length of one but retains its direction.For example, in the original solution:- The gradient vector \( abla f \) at point \( (1,2) \) is \( (-1, -2) \).- Magnitude \( \| abla f \| = \sqrt{(-1)^2 + (-2)^2} = \sqrt{5} \).- The unit vector \( \mathbf{u} \) in this direction is \( \left( \frac{-1}{\sqrt{5}}, \frac{-2}{\sqrt{5}} \right) \).Using a unit vector lets us conveniently express directions without altering the original problem's scale or magnitude of change.