Final answer:
To calculate the gradient of the function h(x,y,z) = xyz^5, one must find the partial derivatives with respect to x, y, and z, resulting in the gradient vector (yz^5, xz^5, 5xyz^4).
Step-by-step explanation:
The question asks to calculate the gradient of the function h(x,y,z) = xyz5. The gradient of a function is a vector that points in the direction of the greatest rate of increase of the function and its magnitude is the slope of the graph in that direction. To find the gradient of h(x,y,z), we need to take the partial derivatives of the function with respect to each variable.To calculate the gradient of the function h(x,y,z) = xyz^5, we need to find the partial derivatives of the function with respect to each variable (x, y, and z).
The partial derivatives are:
∂h/∂x = yz^5
∂h/∂y = xz^5
∂h/∂z = 5xyz^4
Therefore, the gradient of the function is (∂h/∂x, ∂h/∂y, ∂h/∂z) = (yz^5, xz^5, 5xyz^4).
The partial derivative of h(x,y,z) with respect to x is yz5, with respect to y is xz5, and with respect to z is 5xyz4. Thus, the gradient vector of the function is given by grad h(x,y,z) = (yz5, xz5, 5xyz4). This gradient can then be evaluated at any point of interest to find the slope at that point.