To find the derivative matrix of the composition of functions f∘g, we need to compute the partial derivatives of f with respect to u and v, and then evaluate them at the point (u(x, y), v(x, y)). Let's calculate the partial derivatives first:
∂f/∂u = cos(u)cos(v)
∂f/∂v = -sin(u)sin(v)
Now, let's substitute u = 4x^2 - 5y and v = 3x - 5y into the partial derivatives:
∂f/∂u = cos((4x^2 - 5y))cos((3x - 5y))
∂f/∂v = -sin((4x^2 - 5y))sin((3x - 5y))
The derivative matrix D(f∘g)(x, y) is a 1x2 matrix (a row vector) where each entry represents the partial derivative of f∘g with respect to x and y, respectively.
D(f∘g)(x, y) = (∂f/∂u ∂f/∂v) evaluated at (u(x, y), v(x, y))
D(f∘g)(x, y) = (cos((4x^2 - 5y))cos((3x - 5y)), -sin((4x^2 - 5y))sin((3x - 5y)))