Hi all. I've been having problems with the question below.
find
I've tried two differentiation methods:
First
which generally gives me:
I'm not 100% sure about that though. Especially differentiating (the denominator of )
Secondly
I dont know the name of this differentiation method or how to describe it, but essentially you differentiate as normal - with respect to x, treating y as a constant.
This time, however, also add whenever you encounter a . Or is it whenever you differentiate ? Or is it when is left from differentiating - as in was a coefficient of ? As you can see this is where my confusion comes in.
If anyone can confirm or help me with this I would appreciate it. Thanks.
WAIT!!!
We're using the theorem wrong, lol. I'm surprised no one has said anything. I just realized this.
NOW you can apply the implicit function theorem. Always move everything over to one side. The expression we obtained earlier would be true only if .
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Your second method would work too. It is just a little more involved.
I will write since z must be treated as a function of x. So we could go through the following.
Treat as a constant, and remember that z is a function. I would write the first term as where . Then calculate remembering that is a function, so apply the product rule to get . So the term is . You could do something similar with the second term.
Adding the terms gives:
.