91Ó°ÊÓ

The gradient is a vector that is crucial in calculating the directional derivative.It is represented by the symbol \( abla f \) and comprises the partial derivatives of a function.For the function \( f(x, y, z) = xy + z^2 \), the gradient is derived from:

• The partial derivative with respect to \( x \), which is \( \frac{\partial f}{\partial x} = y \).
• The partial derivative with respect to \( y \), which is \( \frac{\partial f}{\partial y} = x \).
• The partial derivative with respect to \( z \), which is \( \frac{\partial f}{\partial z} = 2z \).

These derivatives are combined to form the gradient vector \( abla f = (y, x, 2z) \).This gradient shows the direction of the steepest increase of the function at any point.
Calculating the gradient at a specific point, such as (1, 1, 1), involves substituting these coordinates into the gradient formula, giving \( abla f(1,1,1) = (1, 1, 2) \).This specific vector can then be used to examine how the function changes at this point.

A unit vector is essential in directional derivatives since it defines the specific direction with magnitude one.To find a unit vector for a given direction, you first need the direction vector.In our exercise, the direction vector points from (1, 1, 1) to (5, -3, 3), calculated as \( (4, -4, 2) \).
Next, you normalize this vector to have a magnitude of one.The magnitude \( \|d\| \) of vector \( (4, -4, 2) \) is calculated as \( \sqrt{4^2 + (-4)^2 + 2^2} = 6 \).Then, each component of the direction vector is divided by this magnitude:

• \( \frac{4}{6} = \frac{2}{3} \)
• \( \frac{-4}{6} = \frac{-2}{3} \)
• \( \frac{2}{6} = \frac{1}{3} \)

This results in the unit vector \( \left( \frac{2}{3}, \frac{-2}{3}, \frac{1}{3} \right) \).Finally, this unit vector is used along with the gradient to find the directional derivative, ensuring the derivative is calculated accurately in the precise direction specified.

Partial derivatives are vital for understanding how a multivariable function changes when only one variable is varied, while others are held constant.
In this case, the function is \( f(x, y, z) = xy + z^2 \).Partial derivatives with respect to \( x, y, \) and \( z \) are calculated separately to form the gradient.

• The partial derivative with respect to \( x \), \( \frac{\partial f}{\partial x} = y \), shows how the function shifts when \( x \) is altered while \( y \) and \( z \) stay constant.
• Similarly, \( \frac{\partial f}{\partial y} = x \) indicates change when \( y \) moves, holding \( x \) and \( z \) steady.
• Finally, \( \frac{\partial f}{\partial z} = 2z \) describes how \( f \) varies as \( z \) changes, independent of \( x \) and \( y \).

These individual aspects of change are fundamental to characterizing the behavior of multivariable functions, helping us understand their overall tendency to increase or decrease as conditions vary.