INSTRUCTOR: Find the gradient of f of x comma y comma z, which is equal to a fraction, the numerator of x squared minus 3y squared plus z squared, and a denominator of 2x plus y minus 4z.

Now, using the fact that we know the quotient rule exists and we can find the partial derivative with respect to x, y, and z of this function, the gradient is found by finding f sub x, the partial of f with respect to x, that is, the partial of f with respect to y, and the partial of f with respect to z.

So f sub x, our partial with respect to x, using our quotient rule and some algebra, would be equal to 2x squared plus 2xy minus 8xz plus 6y squared minus 2z squared. All of that is a numerator, with a denominator of, open parentheses, 2x plus y minus 4z, close parentheses, squared.

Similarly, f sub y, the partial of f with respect to y, would be equal to a fraction with a numerator of negative 12xy minus 3y squared plus 24yz minus x squared minus z squared, and a denominator of, open parentheses, 2x plus y minus 4z, close parentheses, squared.

And finally, f sub z, the partial of f with respect to z. So it would be equal to a fraction with a numerator of 4xz plus 2yz minus 4z squared plus 4x squared minus 12y squared, and a denominator of, open parentheses, 2x plus y minus 4z, close parentheses, squared.

Now, each of these partial derivatives we've just found will be the i, j, and k components, respectively, so that the gradient of f of x comma y comma z is equal to a fraction of the numerator of 2x squared plus 2xy minus 8xz plus 6y squared minus 2z squared, and a denominator of, open parentheses, 2x plus y minus 4z, close parentheses, squared, end fraction, i plus a fraction with a numerator of negative 12xy minus 3y squared plus 24yz minus x squared minus z squared, and a denominator of, open parentheses, 2x plus y minus 4z, close parentheses, squared, end fraction, j plus a fraction with a numerator of 4xz plus 2yz minus 4z squared plus 4x squared minus 12y squared, with a denominator of, open parentheses, 2x plus y minus 4z, close parentheses, squared, end fraction, k.