Question

Let f(x, y) = 3x²y + xe^y - x/y.

(a) Find the partial derivatives fₓ and fᵧ.

(b) Compute the directional derivative of f, Dᵤf, in the direction of vector u.

Answer 1

The partial derivatives of the function f(x, y) = 3x²y + xe˥ - x/y are fₓ = 6xy + e˥ - 1/y and fₒ = 3x² + xe˥ + x/y². To compute the directional derivative Dₒf in the direction of a vector u, calculate the gradient ∇f, normalize u to get the unit vector š, and compute the scalar product of ∇f and š.

Step-by-step explanation:

To find the partial derivatives fₓ (partial derivative of f with respect to x) and fₒ (partial derivative of f with respect to y) of the function f(x, y) = 3x²y + xe˥ - x/y, we perform the following calculations:

• For fₓ, we treat y as a constant and differentiate with respect to x:
  fₓ = ∂f/∂x = 6xy + e˥ - 1/y.
• For fₒ, we treat x as a constant and differentiate with respect to y:
  fₒ = ∂f/∂y = 3x² + xe˥(1) - (-x/y²) = 3x² + xe˥ + x/y².

To compute the directional derivative Dₒf of f at a point (say (x_0, y_0)) in the direction of a vector u, we use the following steps:

1. Calculate the gradient of f at (x_0, y_0), ∇f = (fₓ, fₒ).
2. Normalize the direction vector u to get the unit vector š.
3. Compute the scalar (dot) product of ∇f and š to get Dₒf.

So, Dₒf (x_0, y_0) = ∇f ⋅ š at the point (x_0, y_0).