Directional Derivative

Rate of change of a multivariable function in the direction of the unit vector .

Directional derivative of in the direction is:

If ‘s first partials and are continuous, then has a directional derivative in any direction .

Also the directional derivative can be written as:

Gradient

Denoted by .

Maxmimum of Directional Derivative

Maximum value of the directional derivative is and occurs when the gradient vector and has the same direction.

Tangent planes to level surfaces

Let

• is a 3-variable function
• is the level surface of at
• be a continuous curve on
• be a point on

All points on satisfy the equation . The below equation can be deduced by differentiating the equation.

The gradient vector of at is perpendicular to the tangent vector of . If , then the tangent plane to at is the one that passes through and has a normal vector .

The equation of the tangent plane:

Normal line

Normal line to at is the line passing through and perpendicular to the tangent plane. The equation of the normal line is: