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Multivariable calculus is an extension of single-variable calculus, which takes into account functions that depend on more than one variable. In this realm, instead of dealing with simple curves, we often examine surfaces or hypersurfaces. Here, a function could take the form \( f(x, y) \), representing a surface in three-dimensional space.
To analyze these functions, we consider how changes in input variables affect the output of the function. This is where concepts like gradients, partial derivatives, and directional derivatives become crucial tools.
Applications of multivariable calculus are vast, ranging from physics and engineering to economics and biological sciences. It enables us to model systems with many interacting variables, providing deeper insights into intricate real-world phenomena.

In multivariable calculus, a partial derivative measures how a function changes as one of its variables changes while keeping the other variables constant. For a function \( f(x, y) \), the partial derivatives with respect to \( x \) and \( y \) are denoted as \( f_x \) and \( f_y \), respectively.
Calculating partial derivatives involves treating all but one variable as constants, similar to calculating derivatives in single-variable calculus. This technique helps us understand how each variable individually influences the overall function.
For example, if \( f(x, y) = \frac{y^2}{x} \), the partial derivative with respect to \( x \) is \( f_x = -\frac{y^2}{x^2} \) and with respect to \( y \) is \( f_y = \frac{2y}{x} \). These results give us the rate at which \( f \) changes concerning each variable independently.

In calculus, the rate of change of a function describes how the output value of the function shifts as the input value varies. In the multivariable context, it can be more complex due to multiple inputs. To simplify understanding, we often use tools like the gradient to evaluate this concept.
The gradient, a vector, provides us with the direction and rate of the steepest ascent of the function. In simpler terms, it tells us which direction to move in, at a particular point, to increase the function's value most rapidly.
By evaluating the gradient at a specific point, as seen in the solution where the gradient at \( P(1, 2) \) is \( (-4, 4) \), we obtain the rate of change in both the \( x \) and \( y \) directions. This vector gives a convenient snapshot of the function's behavior around the point.

The concept of the directional derivative extends the idea of partial derivatives and allows us to find the rate of change of a function in any specified direction. It combines the gradient vector with a unit vector in the desired direction.
The formula for the directional derivative of a function \( f \) at a point \( P \) in direction \( \mathbf{u} \) is the dot product \( abla f(P) \cdot \mathbf{u} \). This computation gives the rate of change along the line in the preferred direction.
In the given problem, finding the directional derivative of \( f \) at point \( P \) in the direction of vector \( \mathbf{u} = \left( \frac{2}{3}, \frac{1}{3}\sqrt{5} \right) \) involved computing \( (-4, 4) \cdot \mathbf{u} \). The result, \( \frac{4\sqrt{5} - 8}{3} \), signifies how quickly the function's value changes in that specific direction.