Differentiating y with Respect to ... y?
Date: 11/05/2010 at 13:37:45 From: jason Subject: Diff y^5 with respect to y^2 In any normal differentiation question, you take the derivative of y with respect to x: y' = f'(x) = [something involving x] But the problem I have is to differentiate y^5 with respect to y^2 -- both the same variable. I thought I could follow the same pattern, starting with f(x) = y^2 Then replace f(x) with f(y^5), and we get (y^5)^2, which simplifies to y^10. The first deriviative of that is 10y^9. That doesn't look like the official answer, though, which the textbook says is 5/2(y^3). How did they come up with that?
Date: 11/05/2010 at 13:59:05 From: Doctor Ali Subject: Re: Diff y^5 with respect to y^2 Hi Jason! Thanks for writing to Dr. Math. Your approach was correct, but you made a tiny mistake that wound up making a big difference. Start with a substitution, as you did -- but introduce a new variable to keep things clear: u = y^2 Notice that we'll then have d d ------ (y^5) = ---- ( u^(5/2) ) d(y^2) du As you know, that will be 5/2 u^(3/2) And now if you substitute y^2 back in for u, you'll get to the answer. Does that make sense? If you are interested in learning more about this, take a look at http://en.wikipedia.org/wiki/Total_derivative That would help you get a better understanding of these problems. Please write back if you still have any difficulties. - Doctor Ali, The Math Forum http://mathforum.org/dr.math/
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