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Math Help - Finding this derivative

  1. #1
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    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.
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  2. #2
    MHF Contributor
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    Hi

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

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

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

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

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

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

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

    -4\frac{t-54}{3t^{\frac{2}{3}}}
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  3. #3
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    Ahhhh. It's pretty obvious now that I see it, haha. Thanks a bunch
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