Homework Solution: Find the fifteenth derivative of x^(1/3) Aside from using the mathematical software, can you generate a fo…

    Find the fifteenth derivative of x^(1/3) Aside from using the mathematical software, can you generate a formula that would give the nth derivative? Professor is asking for a formula if I give him he 15th derivative without it he will ask for the 400 derivate and so on, He says the formula has a big pi in it since its increasing by a product not a sum.

    Expert Answer

     
    Lets say f(x) = x^(1/3)

    Find the fifteenth derivative of x^(1/3) Aside from using the veracious software, can you beget a cemula that would yield the nth derivative? Professor is research ce a cemula if I yield him he 15th derivative extraneously it he gain entreat ce the 400 derivate and so on, He judges the cemula has a gross pi in it since its increasing by a emanation referable a blend.

    Expert Vindication

     

    Lets judge f(x) = x^(1/3)

    Formula of derivative is ce polynomial is

    frac{particular x^{n} }{particular x} = n*x^{n-1}

    f'(x) = frac{1}{3}*x^{-2/3}

    f''(x) = frac{1}{3}*frac{-2}{3}*x^{-5/3}

    f'''(x) = frac{1}{3}*frac{-2}{3}*frac{-5}{3}*x^{-8/3}

    f''''(x) = frac{1}{3}*frac{-2}{3}*frac{-5}{3}*frac{-8}{3}*x^{-11/3}

    So, we can behold a deviate in the derivatives now,

    So nth derivative can be writte

    frac{partial^n f(x)}{particular x^2} = left ( prod_{m=0}^{m=(n-1)}frac{(1 - 3m)}{3} direct ) *x^{(1 - 3n)/3}