Question

Calculate the gradient of h(x, y, z) = x - 2y^3/ (z). ∇h =

A) ∂h/∂x = 1, ∂h/∂y = -6y^2/z, ∂h/∂z = 2y^3/z^2
B) ∂h/∂x = 1, ∂h/∂y = -6y^3/z^2, ∂h/∂z = 2y^3/z^2
C) ∂h/∂x = 1, ∂h/∂y = -6y^3/z^3, ∂h/∂z = 2y^3/z^2
D) ∂h/∂x = 1, ∂h/∂y = -6y^2/z^2, ∂h/∂z = 2y^3/z^3

Answer 1

The gradient of the function h(x, y, z) = x - 2y^3/z can be found by taking the partial derivatives of each variable. The correct option is A) ∂h/∂x = 1, ∂h/∂y = -6y^2/z, ∂h/∂z = 2y^3/z^2.

Step-by-step explanation:

The gradient of the function h(x, y, z) = x - 2y^3/z is calculated by taking the partial derivative of each variable, x, y, and z.

To find ∂h/∂x, the derivative of x is 1 since x is not dependent on any other variables.

To find ∂h/∂y, the derivative of -2y^3/z is -6y^2/z since y is dependent on the function and z is not.

To find ∂h/∂z, the derivative of -2y^3/z is 2y^3/z^2 since z is dependent on the function and y is not.

Therefore, the correct option is A) ∂h/∂x = 1, ∂h/∂y = -6y^2/z, ∂h/∂z = 2y^3/z^2.