1. Sec 15.6 Directional Derivatives and the Gradient Vector
Definition: Let f be a function of two variables.
The directional derivative of f at in the direction of a unit vector is
if this limit exists.

2. Definition: The Gradient Vector
Theorem:
If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector
and
Definition: The Gradient Vector
If f is a function of x and y, then the gradient of f (denoted by grad f or ) is defined by
Note:

3. The gradient vector (grad f or ) is
Theorem:
If f (x, y, z) is differentiable and , , then the directional derivative is
The gradient vector (grad f or ) is
And

4. Maximizing the Directional Derivative
Theorem: Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative is and it occurs when u has the same direction as the gradient vector

5. Tangent Planes to Level Surfaces
Definition
The tangent plane to the level surface F(x, y, z) = k at the point is the plane that passes through P and has normal vector
Theorem:
The equation of the tangent plane to the level surface
F(x, y, z) = k at the point is

6. Definition
The normal line to a surface S at the point P is the line passing through P and perpendicular to the tangent plane. Its direction is the gradient vector at P.
Theorem:
The equation of the normal line to the level surface
F(x, y, z) = k at the point is