Question

Find the directional derivative of f at point P in the direction of vector v.

Answer 1

The directional derivative of a function f at point P in the direction of vector v is given by the dot product of the gradient of f at point P and the unit vector in the direction of vector v.

Step-by-step explanation:

The directional derivative of a function f at a point P in the direction of a vector v is given by the dot product of the gradient of f at P and the unit vector in the direction of v. The formula for the directional derivative is:

`\[ D_v f(P) = \\abla f(P) \cdot (v)/(\|v\|) \]`

where
`\( \\abla f(P) \)` is the gradient of f at P and
`\( (v)/(\|v\|) \)` is the unit vector in the direction of v.

The directional derivative measures the rate of change of the function f at the point P in the direction of the vector v. It tells us how fast the function is changing in the direction of v at the point P.

To find the directional derivative of a function at a point in the direction of a vector, we first need to find the gradient of the function at the point. Then we take the dot product of the gradient and the unit vector in the direction of the vector. This gives us the directional derivative of the function at the point in the direction of the vector.