INSTRUCTOR: Calculate the partial of f, with respect to x, and the partial of f, with respect to y, for the function f of xy equals tangent of parentheses x cubed minus 3x squared y squared plus 2y to the 4th, close parentheses, by holding the opposite variable constant, then differentiating.

Because this is a composite function, the partial of f with respect to x, first of all, is taking the derivative of tangent. Now, because this is x and y, this will be the same for both parts of this question. The derivative of tangent is secant squared. So we'll have secant squared of that same quantity, x cubed minus 3x squared y squared plus 2y to the 4th, close parentheses, multiplied by the partial with respect to x of x cubed minus 3x squared y squared plus 2y to the 4th, which is the inside function there.

Now, the way that this works is take the derivative of x, as we typically would, but we treat y as if it is a constant, as if it's a coefficient, in a lot of cases. So the derivative of x cubed is 3x squared, but the derivative of 3x squared y squared will be 3y squared times 2x. The derivative of x squared is 2x.

So this means we have minus 6y squared x. And then 2y to the fourth, the derivative of that, with respect to x, treating y as if it's a constant, the derivative of that is 0.

So our final answer for this is secant squared parentheses x cubed minus 3x squared y squared plus 2y to the 4th, close parentheses, open parentheses 3x squared minus 6y squared x.

Now, for the partial of f with respect to y, same process. The derivative of tangent is secant squared of parentheses x cubed minus 3x squared y squared plus 2y to the 4th, close parentheses. But now we'll take the partial, with respect to y, of x cubed minus 3x squared y squared plus 2y to the fourth.

So now x is a constant, not y. So the derivative of x cubed is 0. The derivative of minus 3x squared y squared, treating x if it's a constant, will be minus 3x squared multiplied by the derivative of y squared, which is 2y. And then plus the derivative of 2y to the fourth, which will be 8y cubed.

So that our final answer becomes secant squared of parentheses x cubed minus 3x squared y squared plus 2y to the 4th, close parentheses. Open parentheses, minus 6x squared y plus 8y cubed, close parentheses.