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The gradient of a function is one of the most crucial concepts when dealing with multivariable functions. It plays a similar role as the derivative for single-variable functions. The gradient of a function, often denoted as \( abla f(x, y) \), is a vector that consists of all the partial derivatives of the function. It points in the direction of the greatest rate of increase of the function.
For a function \( f(x, y) \), the gradient is given by:

• \( abla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \)

In simpler terms, whichever way this vector points, the function increases most rapidly. Calculating the gradient helps us understand how a change in one variable affects the entire function. If you're standing on a mountainside, the gradient vector points directly uphill.

Partial derivatives are like the regular derivatives you learned in basic calculus but are applied to functions with more than one variable. They help determine how a function changes as one particular variable changes, holding all other variables constant.
When you compute the partial derivative of a function with respect to \( x \), you differentiate as if \( x \) is the only variable, treating all other variables as constants:

• \( \frac{\partial f}{\partial x} \)

Similarly, for Partial derivative with respect to \( y \):

• \( \frac{\partial f}{\partial y} \)

These derivatives are foundational for creating the gradient and are fundamental in calculating the directional derivative, which tells us the rate of change of our function in any given direction.

A unit vector is a vector with a magnitude of 1. It's used to indicate direction without altering length. Unit vectors are particularly useful in directional derivatives because they provide a standardized way to express a direction, ensuring the derivative measures the rate of change purely in terms of direction and not magnitude.
To find a unit vector in a given direction described by an angle \( \theta \):

• \( \mathbf{u} = (\cos \theta, \sin \theta) \)

In our exercise, \( \theta = \frac{\pi}{4} \) gives a unit vector \( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \). This vector orientation ensures the uniformity needed to compute directional derivatives, making it possible to explore how a function changes not just at a point, but in a specific direction.

The dot product is a fundamental operation when working with vectors. It's a method of multiplying two vectors that results in a scalar. In the context of directional derivatives, the dot product measures how much of one vector 'goes in the direction' of another.
When calculating a directional derivative, you compute the dot product between the gradient and the unit vector in the desired direction:

• \( D_{\mathbf{u}}f = abla f \cdot \mathbf{u} \)

The formula simplifies in our example to:

• \( 5 \cdot \frac{\sqrt{2}}{2} + (-11) \cdot \frac{\sqrt{2}}{2} \)

The resulting value indicates how quickly the function increases or decreases in the specified direction. If the dot product is positive, the function increases; if negative, it decreases in that direction.