Final answer:
The partial derivatives of the function f(x, y, z) = 1 + 8xy^3 - 5z^4 with respect to x, y, and z are f_x = 8y^3, f_y = 24xy^2, and f_z = -20z^3 respectively. These represent the rates of change of the function along the respective axes.
Step-by-step explanation:
To find f_{x}, f_{y}, and f_{z} for the function f(x, y, z) = 1 + 8xy^3 - 5z^4, we need to take the partial derivatives of the function with respect to each variable. The partial derivative with respect to x is f_{x} = ∂8y^3, which represents the rate of change of f in the direction of the x-axis.
The partial derivative with respect to y is f_{y} = 24xy^2, representing the rate of change of f in the direction of the y-axis. Lastly, the partial derivative with respect to z is f_{z} = -20z^3, showing the rate of change of f in the direction of the z-axis.