Find the partial derivatives indicated Assume the variables are restricted to a domain on which the function is defined. z=x8+3y+xy.

Answer: To find the partial derivatives of the function z = x^8 + 3y + xy with respect to x and y, we use the following approach:

1. Partial derivative of z with respect to x (∂z/∂x): Treat y as a constant, and differentiate the function z with respect to x. - The derivative of x^8 with respect to x is 8x^7 (using the power rule). - The derivative of the constant term 3y with respect to x is 0 (since it does not involve x). - The derivative of xy with respect to x is y (finding the derivative of the first function x with respect to x, which is 1, and then multiplying by the second function y which is treated as a constant). Putting it all together: ∂z/∂x = 8x^7 + 0 + y = 8x^7 + y.

2. Partial derivative of z with respect to y (∂z/∂y): Treat x as a constant, and differentiate the function z with respect to y. - The derivative of the constant term x^8 with respect to y is 0 (since it does not involve y). - The derivative of 3y with respect to y is 3 (using the constant multiple rule). - The derivative of xy with respect to y is x (finding the derivative of the second function y with respect to y, which is 1, and then multiplying by the first function x which is treated as a constant). Putting it all together: ∂z/∂y = 0 + 3 + x = 3 + x.

Therefore, the partial derivatives of the function z are: ∂z/∂x = 8x^7 + y ∂z/∂y = 3 + x .