Final answer:
The method to find the total differential for z = xy + y^x involves partial derivatives and changes in x and y, but precise computations cannot be made without actual values for the partial derivatives with the information provided.
Step-by-step explanation:
The total differential (dz) of the function z = xy + y^x, when x changes from 10 to 10.5 and y changes from 15 to 13 is found using the method of partial derivatives. First, calculate the partial derivatives dz/dx and dz/dy, then multiply them by the changes in x and y, respectively.
To calculate the partial derivatives for z = xy + y^x (where y^x is interpreted as y raised to the power of x):
dz/dx = y + y^xln(y), since the derivative of y^x with respect to x is y^xln(y).
dz/dy = x + xy^(x-1), since the derivative of xy with respect to y is x, and the derivative of y^x is xy^(x-1).
Applying the changes, Δx = 10.5 - 10 = 0.5 and Δy = 13 - 15 = -2:
dz = (dz/dx)Δx + (dz/dy)Δy
= (15 + 15^10ln(15)) * 0.5 + (10 + 10 * 13^(10-1)) * (-2)
This computation gives us the total differential dz. Without precise values for the partial derivatives and multiplication, the answer options provided are not directly calculable and an exact value for dz cannot be determined from the given information.