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.
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partial derivatives are just like normal ones, except all other variables are treated like constants. For example,
B(x, y) = xe^(xy) − y
Bx = e^(xy) + xye^(xy)
there is no product rule involving e^(xy), since y is just a constant
By = x*xe^(xy) - 1
To calculate the derivatives of A and B with respect to x and y, we can use the rules of differentiation. Here's a step-by-step breakdown of how to find each derivative:
(a) To calculate the x derivative of A, or Ax, we differentiate A(x, y) with respect to x while treating y as a constant. Applying the power rule and product rule, we get:
Ax = (2xy + y - 3y^2)
(b) To calculate the y derivative of A, or Ay, we differentiate A(x, y) with respect to y while treating x as a constant. Applying the product rule, we get:
Ay = (x^2 - 6xy + x)
(c) To calculate the x derivative of B, or Bx, we differentiate B(x, y) with respect to x while treating y as a constant. Applying the product rule and the chain rule (for e^xy), we get:
Bx = (e^xy + xy*e^xy)
(d) To calculate the y derivative of B, or By, we differentiate B(x, y) with respect to y while treating x as a constant. This simplifies the differentiation to:
By = (xe^xy - 1)
By following the rules of differentiation and applying them step by step, we were able to calculate the x and y derivatives of both functions A(x, y) and B(x, y).