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/