INSTRUCTOR: Calculate the partial of w with respect to u and the partial of w with respect to v given the following functions. w equal to f of x comma y comma z is equal to numerator x plus 2y minus 4z, with a denominator of 2x minus y plus 3z, x equal to x of u comma v, which is equal to e to the 2u, end exponent, cosine, parentheses, 3v, close parentheses, y equal to y of u comma v equal to e to the 2u, end exponent, sine, open parentheses, 3v, close parentheses, and z equal to z of u comma v, which is e to the 2u.

Based on our chain rule for three variables, we need to find partial derivatives of all of these functions, with respect to each of their independent variables. So I'm going to show a little of my work, but not all of it. Some of this is scratch work. So w sub x, which is the partial of w with respect to x-- I'll often say this as wx-- using our quotient rule, that is equal to a numerator of negative 5y plus 11z with a denominator of, parentheses, 2x minus y plus 3z, close parentheses, squared.

Now, knowing the values of x, y, and z, they are functions, we see that this is going to be equal to a numerator of negative 5 sine, open parentheses, 3v, close parentheses, plus 11, and a denominator of e to the 2u, end exponent, open parentheses, 2 cosine, open parentheses, 3v, close parentheses, minus sine, open parentheses, 3v, close parentheses, plus 3. And that denominator is all squared there, or that portion of the denominator is squared.

Now, I'll also note that as I was simplifying this, I did factor out an e to the 2u and reduce that. OK, so just be aware of that.

All right, next we have wy, which, using our quotient rule, is equal to a numerator of 5x plus 2z over a denominator of, parentheses, 2x minus y plus 3z, close parentheses, squared.

And again, making our substitutions, that will be equal to a numerator of 5 cosine, open parentheses, 3v, close parentheses, plus 2, with a denominator of e to the 2u, end exponent, open parentheses, 2 cosine, open parentheses, 3v, close parentheses, minus sine, open parentheses, 3v, close parentheses, plus 3, close parentheses, squared.

Finally, wz, with our quotient rule, is equal to a numerator of negative 11x minus 2y and a denominator of, open parentheses, 2x minus y plus 3z, close parentheses, squared.

And that expression will be equal to a numerator of negative 11 cosine, open parentheses, m close parentheses, minus 2 sine, open parentheses, m close parentheses, and a denominator of e to the 2u, end exponent, open parentheses, to cosine, open parentheses, 3v, close parentheses, minus sine, open parentheses, 3v, close parentheses, plus 3, close parentheses, squared.

Next, we want to find the partials of x with respect to u and v, y with respect to u and v, and z with respect to u and v. So we can apply our chain rule. So xu is equal to 2e to the 2u, end exponent, cosine, open parentheses, 3v, close parentheses.

And then x sub v is equal to negative 3e to the 2u, end exponent, sine, open parentheses, 3v, close parentheses, yu equal to 2e to the 2u, end exponent, sine, open parentheses, 3 of e, close parentheses, yv will be equal to 3e to the 2u, end exponent, cosine, open parentheses, 3v, close parentheses.

And then we have zu, which is equal to 2e to the 2u, end exponent, and zv, which is equal to 0.

Now, for the chain rule itself, we know that w-- partial w with respect to u-- is equal to wx multiplied by xu plus wy multiplied by yu plus wz multiplied by zu, and wv equals wx multiplied by xv plus wy multiplied by yv plus wz multiplied by zv.

Now, we have these expressions. We're going to-- really, we're actually finding a dot product, in essence, as we've seen before, with these two functions, two sets of functions.

All right, now, if we combine functions or we look for like terms and things very carefully, we'll actually notice that we're going to have some terms simplify. That's what we want.

All right, so wu is going to be equal to-- in fact, they all have the same denominator, so we can make the go-- we can make this into one fraction.

We're going to have a numerator of 2 cosine, open parentheses, 3 of e, open parentheses, negative 5 sine, open parentheses, 3v, close parentheses, plus 11 plus 2 sine, open parentheses, 3v, open parentheses, 5 cosine, open parentheses, 3v, close parentheses, plus 2, close parentheses, plus 2, open parentheses, negative 11, cosine, open parentheses, 3v, close parentheses, minus 2 sine, open parentheses, 3v, close parentheses, close parentheses.

That is our numerator over our common denominator of, open parentheses, 2 cosine, open parentheses, 3v, close parentheses, minus sine, open parentheses, 3v, close parentheses, plus 3, close parentheses, squared.

Now, if you are to consider those terms, we have many terms that cancel out. In fact, all of this, so that wu is going to be equal to 0.

Now you're going to follow a similar pattern to see that wv, with some algebra, is equal to a numerator of 15 minus 33 sine, open parentheses, 3v, close parentheses, plus 6 cosine, open parentheses, 3v, close parentheses, with a denominator of, parentheses, 2 cosine, open parentheses, 3v, close parentheses, minus sine, open parentheses, 3v, close parentheses, plus 3, close parentheses, squared.

And that is the derivative of partial of w with respect to u and partial of w with respect to v.