Final answer:
The gradient of the function f(x, y) = cos(x^5 + y) is computed by taking the partial derivatives with respect to x and y, resulting in the vector (-sin(x^5 + y) · 5x^4, -sin(x^5 + y)).
Step-by-step explanation:
To calculate the gradient of the function f(x, y) = cos(x⁵ + y), we need to find the partial derivatives with respect to x and y. The gradient Vf is a vector that contains these partial derivatives as its components. To compute the partial derivative with respect to x, denoted as Fx, we apply the chain rule for derivatives which gives us Fx = -sin(x⁵ + y) · 5x⁴.
Similarly, the partial derivative with respect to y, denoted as Fy, is simply Fy = -sin(x⁵ + y) because the derivative of cos(u) with respect to u is -sin(u) and the derivative of y with respect to y is 1.
The gradient of f(x, y) is then the vector Vf given by (Fx, Fy), which in this case is (-sin(x⁵ + y) · 5x⁴, -sin(x⁵ + y)).
To calculate the gradient of the function f(x, y) = cos(x⁵ + y), we need to find the partial derivatives of f with respect to x and y. The gradient vector is then given by (∂f/∂x, ∂f/∂y). In this case, ∂f/∂x = -5x⁴sin(x⁵ + y) and ∂f/∂y = -sin(x⁵ + y). Therefore, the gradient is (-5x⁴sin(x⁵ + y), -sin(x⁵ + y)).