INSTRUCTOR: Calculate the partial of z with respect to u and the partial of z with respect to v given the following functions. z equals f of x comma y equal to numerator 2x minus y denominator x plus 3y x of u comma v equal to e to the 2u multiplied by cosine parentheses 3v closed parentheses and y of uv equal to e to the exponent of 2u end exponent sine multiplied by sine of parentheses 3v closed parentheses. Now, the chain rule for partials, we'll run into in just a moment. I'll state what that is. But it's going to involve partials of each of these with respect to their respective variables.

So we're going to rely on our knowledge of partial derivatives heavily. So zu, which is what we are trying to find first, by definition is equal to zx multiplied by xu plus zy multiplied by yu. And zv is equal to zx multiplied by xv plus zy multiplied by yv. Notice I'm using the subscripts to indicate partial derivatives. So z sub u is the partial of z with respect to u, et cetera.

So let's find each of these values. Knowing the quotient rule and a little bit of scratch work, I can say that zx is going to be equal to 7y as a numerator divided by parentheses x plus 3y closed parentheses squared as our denominator. And zy, again with the quotient rule, equal to a numerator of negative 7x the denominator of parentheses x plus 3y closed parentheses squared. Now, we know that x and y are functions of u and v So we will substitute those values for those functions into z sub x and z sub y in just a moment.

Now for the partials of x and y with respect to u and v. We'll find x sub u is equal to 2e to the 2. So 2e to the exponent of 2u end exponent multiplied by cosine open parentheses 3v. xv is equal to negative 3e with an exponent of 2u end exponent sine open parentheses 3v closed parentheses. So there are two partials of x, and let's find our two partials of y.

yu will be equal to 2e to the exponent of 2u end exponent sine open parentheses 3v closed parentheses. And y sub v is equal to 3e to the exponent of 2u end exponent cosine open parentheses 3v closed parentheses. Now we can apply our chain rule definition to determine what zu and zv are. And I'm going to simultaneously use that definition as I'm also going to evaluate for my function x of uv and my function y of uv into the z sub x and z sub y. All right.

So keep track of your variables carefully. So the partial z with respect to u is equal to a numerator of 7e to the exponent of 2u end exponent sine 3v end parentheses multiplied by 2e to the exponent of 2u end exponent cosine open parentheses 3v closed parentheses over open parentheses e to the 2u cosine open parentheses 3v closed parentheses plus 3e to the 2u sine open parentheses 3v closed parentheses, close our denominator parentheses, n squared minus, again, I'm substituting my x and y functions here, so I have minus numerator 7e to the 2u end exponent multiplied by cosine open parentheses 3v closed parentheses multiplied by 2e to the 2u end exponent sine open parentheses 3v closed parentheses with a denominator that is exactly the same as the first term open parentheses e to the 2u end exponent cosine parentheses 3v closed parentheses plus 3e to the exponent of 2u end exponent sine open parentheses 3v closed parentheses closed parentheses squared.

Now with some algebra, because these have the same denominator, both in terms of the same denominator, you might notice that these are the exact same numerator written in a different order and we're subtracting them. So the partial of z with respect to u is equal to 0. Now, for the partial with respect to z-- or partial of z with respect to v, we're going to have a very similar fraction. So the partial of z with respect to v will be equal to a numerator of 7e to the 2u end exponent sine open parentheses 3v closed parentheses times negative 3e to the exponent of 2u end exponent multiplied by sine open parentheses 3v closed parentheses over open parentheses e to the 2u end exponent cosine open parentheses 3v closed parentheses plus 3e to the 2u end exponent sine open parentheses 3v closed parentheses closed parentheses squared minus 7e to the 2u end exponent cosine open parentheses 3v closed parentheses multiplied by 3e to the 2u end exponent cosine open parentheses 3v closed parentheses. That was our numerator.

Our denominator is open parentheses e to the 2u end exponent cosine open parentheses 3v closed parentheses plus 3e to the 2u end exponent sine open parentheses 3v closed parentheses closed parentheses squared. Now as we notice what values we have in our two numerators, again, same denominator, everything has an e to the 2u times e to the 2u. We have a 7 times a negative 3 and a negative 7 times 3. All right. So we can factor out colorize-- oh, and we have sine 3v times sine of 3v in the first term and the second term we have cosine of 3v times cosine of 3v.

All right. I'm going to factor this into one term. First, we have a negative 21e to the 4u open parentheses sine squared open parentheses 3v closed parentheses plus cosine squared open parentheses 3v closed parentheses closed parentheses. That's our numerator. Divided by open parentheses e to the 2u end exponent cosine open parentheses 3v closed parentheses plus 3e to the 2u end exponent sine open parentheses 3v closed parentheses closed parentheses squared.

Now we can factor out an e to the 4u because we have an e to the 2u squared in the denominator. We can factor that part out. And also knowing that sine squared plus cosine squared regardless of the arguments is equal to 1. This is going to be equal to a numerator of negative 21 with a denominator of open parentheses cosine open parentheses 3 of e closed parentheses plus 3 sine open parentheses 3v closed parentheses closed parentheses squared.