Consider the functions A(x, y) = x^2y + xy − 3xy^2, and B(x, y) = xe^xy − y.
(a) Calculate the x derivative of A, or Ax.
(b) Calculate the y derivative of A, or Ay.
(c) Calculate the x derivative of B, or Bx.
(d) Calculate the y derivative of B, or By.
To calculate the derivatives of the given functions, we need to use some basic rules of differentiation. The general rule for finding the derivative of a function with respect to a variable is to differentiate each term of the function separately while keeping the other variables constant.
Let's begin with function A(x, y) = x^2y + xy - 3xy^2.
(a) To calculate the x derivative of A, or Ax, we differentiate each term with respect to x while treating y as a constant:
Ax = d/dx(x^2y) + d/dx(xy) - d/dx(3xy^2).
The derivative of x^2 with respect to x is 2x, treating y as a constant. Similarly, the derivative of xy with respect to x is y, treating y as a constant. And the derivative of 3xy^2 with respect to x is 3y^2, treating y as a constant.
So Ax = 2xy + y - 3y^2.
(b) To calculate the y derivative of A, or Ay, we differentiate each term with respect to y while treating x as a constant:
Ay = d/dy(x^2y) + d/dy(xy) - d/dy(3xy^2).
The derivative of x^2y with respect to y is x^2, treating x as a constant. The derivative of xy with respect to y is x, treating x as a constant. And the derivative of 3xy^2 with respect to y is 6xy, treating x as a constant.
So Ay = x^2 + x - 6xy.
(c) Now let's move on to function B(x, y) = xe^xy - y.
To calculate the x derivative of B, or Bx, we differentiate each term with respect to x while treating y as a constant:
Bx = d/dx(xe^xy) - d/dx(y).
The derivative of xe^xy with respect to x can be found using the product rule. Let u = x and v = e^xy. Then, using the product rule: d/dx(xe^xy) = u(dv/dx) + v(du/dx).
For u = x, du/dx = 1 (the derivative of x with respect to x is 1). For v = e^xy, dv/dx = (d/dx)(e^xy) = ye^xy (using the chain rule).
So, d/dx(xe^xy) = x(ye^xy) + e^xy.
The derivative of y with respect to x is 0 since it does not depend on x.
So Bx = x(ye^xy) + e^xy - 0 = xye^xy + e^xy.
(d) To calculate the y derivative of B, or By, we differentiate each term with respect to y while treating x as a constant:
By = d/dy(xe^xy) - d/dy(y).
The derivative of xe^xy with respect to y can be found using the product rule as before. Let u = x and v = e^xy. Then, d/dy(xe^xy) = u(dv/dy) + v(du/dy).
For u = x, du/dy = 0 since x does not depend on y. For v = e^xy, dv/dy = (d/dy)(e^xy) = xe^xy (using the chain rule).
So, d/dy(xe^xy) = x(xe^xy) + e^xy.
The derivative of y with respect to y is 1.
So By = x(xe^xy) + e^xy - 1 = x^2e^xy + e^xy - 1.
That's it! We have now obtained the x and y derivatives of the given functions.