INSTRUCTOR: Calculate the partial of f with respect to x, the partial of f with respect to y, and the partial of f with respect to z for the function f of x comma y comma z equal to secant, parentheses, x squared times y, close parentheses, minus tangent, open parentheses, x cubed times y times z squared, close parentheses.

In order to find each of these partials, it's going to be necessary to talk about what it means to find the partial with respect to x of something like x squared y. So let's walk through that.

So the partial with respect to x of x squared y means that we're going to treat every variable that is not an x as a constant, and factor it out as such. So this will be equal to y times the derivative of x squared, which would be equal to 2xy.

Similarly, the partial with respect to y of x squared times y, for that we would treat the x squared as a constant. So this would be the x squared times the derivative with respect to y of y, which would be equal to x squared.

And finding the partial of either of those with respect to z would be equal to 0 because then, in that case, x and y are both constants. On a similar note, see what the partial with respect to x of x cubed times y times z squared would be.

For that, we'd treat y and z squared as constant. So y times z squared multiplied by the derivative with respect to x of x cubed, because the derivative with respect to x of x cubed is 3x squared. This would be 3x squared yz squared.

All right, now we'll find the partial with respect to y of that same expression of x cubed times y times z squared. We'll treat x and z as constants. So this would be x cubed z squared multiplied by the derivative with respect to y of y. That's equal to 1. So that particular partial is x cubed z squared.

And then, finally, the partial with respect to z of x cubed times y times z squared. For that, we would treat x and y as constants. So we'd have x cubed y multiplied by the derivative with respect to z of z squared. That derivative would be 2z. So this partial with respect to z would be 2x cubed yz.

Now we're going to apply the chain rule to our original function f of xyz so that the partial of f with respect to x would be equal to secant of x squared y multiplied by tangent of x squared y multiplied by the partial with respect to x of x squared y, which we already determined is 2xy. So multiply 2xy.

So the partial of f with respect to x is secant, parentheses, x squared times y, close parentheses, tangent, parentheses, x squared times y, close parentheses, multiplied by 2xy minus the derivative of tangent is secant squared. So this would be secant squared, open parentheses, x cubed times y times z squared, close parentheses, multiplied by 3x squared yz squared, using the previous partials that we have.

All right, this makes the partial of f with respect to 2y equal to secant, open parentheses, x squared y, close parentheses, times tangent, open parentheses, x squared y, close parentheses, multiplied by x squared minus secant squared, open parentheses, x cubed yz squared, close parentheses, multiplied by x cubed z squared.

And finally, the partial of f with respect to z is equal to negative secant squared, open parentheses, x cubed yz squared, close parentheses, multiplied by 2x cubed yz.