Calculate all four second-order partial derivatives and confirm that the mixed partials are equal. f(x, y)= 2e^xy

ANSWER TO QUESTION 1

The given function is



The partial derivative of f with respect to x means we are treating y as a constant. The first derivative is



and the second derivative with respect to x is,



ANSWER TO QUESTION 2

The given function is



The partial derivative of f with respect to y means we are treating x as a constant. The first derivative is



and the second derivative with respect to y is



ANSWER TO QUESTION 3

Our first mixed partial is



We need to differentiate

again. But this time with respect to y.

Since this is a product of two functions of y, we apply the product rule of differentiation to obtain,





ANSWER TO QUESTION 4

The second mixed partial is



We need to differentiate


again. But this time with respect to x.

Since this is a product of two functions of x, we apply the product rule of differentiation to obtain,





Hence,