Find the directional derivative of the given function at the...

Find the directional derivative of the given function at the specified point $P$ and in the direction of the given unit vector $\mathbf{u}$.
$$
f(x, y)=\sqrt{\frac{y}{x}} \text { at } P=(4,9), \mathbf{u}=\frac{15}{17} \mathbf{i}+\frac{8}{17} \mathbf{j}
$$

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Key Concepts

Gradient

The gradient of a scalar function is a vector composed of its partial derivatives with respect to each variable. It indicates the direction of the steepest ascent of the function and is central in analyzing how the function changes locally.

Partial Derivatives

Partial derivatives measure the rate of change of the function with respect to one variable at a time while keeping the other variables constant. They are the building blocks for constructing the gradient vector.

Directional Derivative

The directional derivative quantifies the rate at which a function changes at a point in a specified direction. It is computed as the dot product between the gradient of the function and a unit vector in the given direction.

Unit Vector

A unit vector has a magnitude of one and indicates direction only. In the context of directional derivatives, it ensures that the rate of change is measured per unit distance in the specified direction.

Evaluation at a Point

Computing values such as the gradient or directional derivative at a specific point involves substituting the coordinates of that point into the derived expressions. This step is crucial for obtaining numerical results related to the behavior of the function at that point.

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