Question

What is the directional derivative of f(x, y)=ln(x²+y²)+x² y² in the direction of the vector v=⟨-3,4⟩ at the point (1,-1) ?

Answer 1

The directional derivative of f(x, y)=ln(x²+y²)+x²y² in the direction of the vector v=⟨-3,4⟩ at the point (1,-1) is equal to the dot product of the gradient of f and the unit vector in the direction of v.

Step-by-step explanation:

The directional derivative of a function f(x, y) in the direction of a vector v at a point (a, b) is given by the dot product of the gradient of f with the unit vector in the direction of v.

In this case, the function is f(x, y) = ln(x²+y²) + x²y², the point is (1, -1), and the vector is v = <-3, 4>.

First, we need to calculate the gradient of f, which is ∇f = (∂f/∂x, ∂f/∂y) = (2x/(x²+y²) + 2x²y², 2y/(x²+y²) + 2x²y²).

Next, we normalize v to get the unit vector u = v/||v|| = <-3/5, 4/5>.

Finally, we calculate the directional derivative by taking the dot product of ∇f and u: Df(u) = ∇f · u = (∂f/∂x, ∂f/∂y) · (-3/5, 4/5).