Final answer:
The gradient of the function h(x, y, z) = x² / (1 / y²) / (1 / z³) can be calculated using partial derivatives. The gradient is given by ∇h = (2xy²z³)i - (2x²z³/y)j - (3x²y²/z⁴)k
Step-by-step explanation:
The gradient of the function h(x, y, z) = x² / (1 / y²) / (1 / z³) can be calculated using partial derivatives. To find the gradient, we take the partial derivative of the function with respect to each variable. Let's calculate:
∇h = (∂h/∂x)i + (∂h/∂y)j + (∂h/∂z)k
Using the chain rule, we can simplify the partial derivatives:
∂h/∂x = 2x / (1/y² * 1/z³) = 2xy²z³
∂h/∂y = -2x²y / (1/y³ * 1/z³) = -2x²z³/y
∂h/∂z = -3x²y² / (1/y² * 1/z⁴) = -3x²y²/z⁴
Therefore, the gradient of h(x, y, z) is: ∇h = (2xy²z³)i - (2x²z³/y)j - (3x²y²/z⁴)k