# Finding this derivative

• Dec 3rd 2008, 08:00 AM
fattydq
Finding this derivative
So here's the deal, I already have the answer to a derivative related problem I was doing because I gave up and used an online derivative calculator, which usually (because it shows the steps involved) makes me realize where I was going wrong if I'm having trouble finding a derivative. But this one, even now that I have the answer, confuses me. The original function is t^(1/3)*(216-t) and I've gotten as far as (216-T/3t^(2/3))-T^(1/3) which is correct, since it shows it in the steps on said online derivative calculator.

It then immediately goes from that to simplifying it down to - 4(t-54)/3t^(2/3)
I understand that 4 factors out of 216 to get the -54, but because there's no 4 in front of the t in the term, and because the -t^(1/3) seems to just disappear, I don't understand how the final step of this problem was accomplished. I know the answer is right because I'm doing online homework and it was shown as the correct answer.
• Dec 3rd 2008, 08:56 AM
running-gag
Hi

The derivative of $\displaystyle t^{\frac{1}{3}}(216-t)$

is $\displaystyle \frac{216-t}{3t^{\frac{2}{3}}}-t^{\frac{1}{3}}$

Using the same denominator ($\displaystyle {3t^{\frac{2}{3}}}$)

$\displaystyle \frac{216-t}{3t^{\frac{2}{3}}}-\frac{3t^{\frac{2}{3}}t^{\frac{1}{3}}}{3t^{\frac{2 }{3}}}$

Now $\displaystyle t^{\frac{2}{3}}t^{\frac{1}{3}}=t$

It gives
$\displaystyle \frac{216-t}{3t^{\frac{2}{3}}}-\frac{3t}{3t^{\frac{2}{3}}}$

$\displaystyle \frac{216-4t}{3t^{\frac{2}{3}}}$

$\displaystyle -4\frac{t-54}{3t^{\frac{2}{3}}}$
• Dec 3rd 2008, 11:34 AM
fattydq
Ahhhh. It's pretty obvious now that I see it, haha. Thanks a bunch