INSTRUCTOR: Calculate all four second partial derivatives for the function f of xy equals sine, parentheses, 3x minus 2y, close parentheses, plus cosine, parentheses, x plus 4y, close parentheses.

Throughout this question, to find the partial with effect to x, the partial of f with respect to y, partial of f with effect to x twice, fxx, fyx, fxy, and fyy, we're going to make extensive use of our knowledge of the derivatives of trig functions, sine and cosine in this case. Also, what we know about partials of polynomials, and putting that all together with the chain rule.

So first fx will be equal to 3, cosine, open parentheses, 3x minus 2y, close parentheses, minus sine, open parenthesis, x plus 4y, close parentheses. Fy will be equal to negative 2, cosine, open parentheses, 3x minus 2y, close parentheses, minus 4, sine, open parenthesis, x plus 4y, close parentheses.

Now to find fxx, fyx, fxy, and fyy, again, we'll use our derivatives of trig functions, partials of polynomials, and the chain rule. Now so fxx is the partial of x of fx with respect to x again. So that will be equal to negative 9, sine, open parentheses, 3x minus 2y, close parentheses, minus cosine, open parenthesis, x plus 4y, close parentheses.

Fyx, which will be the derivative of fx with respect to y, will be 6, sine, open parentheses, 3x minus 2y, close parentheses, minus 4, cosine, open parenthesis, x plus 4y.

For our next pair, we'll be taking the derivative of fy with respect to x, and then with respect to y. So fxy will be equal to 6, sine, open parentheses, 3x minus 2y, close parentheses, minus 4, cosine, open parenthesis, x plus 4y, close parentheses.

And finally, fyy, again, that is the derivative of fy with respect to y once again. And that will be minus 4, sine, open parentheses, 3x minus 2y, close parentheses, minus 16, cosine, open parenthesis, x plus 4y, close parentheses.