00:01
So in this question, they say, i want to find the partial derivative of u with respect to p using the chain rule if u is equal to the natural log of xy plus e to the power of 3xy squared.
00:16
My x is equal to pq and my y is equal to 3p plus 2q.
00:23
Now, they give you the answer, but the question is, how did they get to that answer? so what do we have here? i have u, which is a function of x and y.
00:37
And then i have x, which is a function of p and q, and i have y, which is also a function of p and q.
00:50
So in order to get my partial derivative of u with respect to p, we're going to need the multivariable chain.
00:59
Now, i remember this by making what's called a tree diagram.
01:04
And that's how most teachers teach this.
01:07
So the idea is i have a function you and that function depends on two independent variables, x and y.
01:18
Now each of x and y depend on p and q.
01:23
So i'm going to draw branches from x and y each to p and q.
01:31
And so if i want the partial derivative of u with respect to p what i do is i look at all the paths on this tree diagram that lead from u to p and i multiply down each step so i take my partial derivative of u with respect to x times the partial derivative of x with respect to p and then i add each of the branches so i multiply down each branch and then i add the result along each branch.
02:10
So i'm getting, for my second branch, i multiply down the partial of u with respect to y times the partial of y with respect to p.
02:22
Now, if i wanted my partial derivative of u with respect to x, what am i getting? well, i get 1 over xy times y by the chain, plus e to the power of 3xy squared times the derivative of that exponent with respect to x, which is 3y squared...