Final answer:
The second partial derivatives of the function v = xy are 0 with respect to both x and y (d2v/dx2 and d2v/dy2) and 1 for the mixed partial derivatives (d2v/dxdy or d2v/dydx).
Step-by-step explanation:
The student has presented a function v = xy and has asked to find all the second partial derivatives of this function. The function they've written seems to have a typo or missing terms, as 'x - y' doesn't appear to be integrated into the function meaningfully. Assuming the function intended was v = xy (given the omission of alternative instructions for the 'x - y' part), the second partial derivatives would be calculated with respect to x and y, denoted as d2v/dx2, d2v/dy2, and the mixed partial derivative d2v/dxdy or d2v/dydx.
Here are the steps for each:
- The second partial derivative of v with respect to x is obtained by first taking the partial derivative of v with respect to x, which gives us y, and then taking the derivative of that with respect to x, which will be 0 since y is a constant with respect to x.
- The second partial derivative of v with respect to y is derived similarly, first finding that the partial derivative of v with respect to y is x, and then differentiating that with respect to y, yielding 0 as x is a constant with respect to y.
- The mixed partial derivatives involve first differentiating with respect to one variable and then the other. In this case, either order of differentiation gives us the derivative of x with respect to y or vice versa, resulting in 1.
Therefore, the second partial derivatives of the function v = xy are:
- d2v/dx2 = 0
- d2v/dy2 = 0
- d2v/dxdy or d2v/dydx = 1