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Math Help - Smart way of finding 4th derivative of function?

  1. #1
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    Smart way of finding 4th derivative of function?

    Is there some smart way of finding the fourth derivative of

    $f(t)=(q+pe^t)^n$,

    perhaps expanding using the binomial theorem?
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  2. #2
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    Re: Smart way of finding 4th derivative of function?

    Just differentiate it (four times), function of a function/product rule.
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  3. #3
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    Re: Smart way of finding 4th derivative of function?

    Quote Originally Posted by MagisterMan View Post
    Is there some smart way of finding the fourth derivative of
    $f(t)=(q+pe^t)^n$,
    You can see several ways here.
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  4. #4
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    Re: Smart way of finding 4th derivative of function?

    Quote Originally Posted by MagisterMan View Post
    Is there some smart way of finding the fourth derivative of

    $f(t)=(q+pe^t)^n$,

    perhaps expanding using the binomial theorem?
    Perhaps you would consider the following smart
    Let x = q + pe^t \Rightarrow x' = pe^t = x-q= x'' = x'''=x''''
    f = x^n
    f' = n x^{n-1} x' =u x'
    f''= (u+u')x'
    f''' = (u+2u'+u'')x'
    f'''' = (u+3u' + 3u'' + u''')x' = ( n x^{n-1} + 3 n (n-1) x^{n-2} + 3 n (n-1) (n-2) x^{n-3} +n(n-1)(n-2) (n-3) x^{n-4} ) (x-q)
    =nx^{n-4} (x^3 + 3(n-1) x^2 + 3 (n-1) (n-2) x + (n-1)(n-2)(n-3) )(x-q)
    Thanks from topsquark and MagisterMan
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