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Math Help - please remind how to do - simple differentiation question

  1. #1
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    please remind how to do - simple differentiation question

    differentiate

    a/1+a

    I just cant remember how to do it
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  2. #2
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    \displaystyle \frac{d}{da}\left(\frac{a}{1+a}\right) = \frac{(1+a)\frac{d}{da}(a) - a\,\frac{d}{da}(1+a)}{(1+a)^2}

    \displaystyle = \frac{(1+a)1 - a(1)}{(1+a)^2}

    \displaystyle = \frac{1+ a - a}{(1 + a)^2}

    \displaystyle = \frac{1}{(1+a)^2}.
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  3. #3
    Senior Member BAdhi's Avatar
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     f(a) = 1 - \frac{1}{a+1}

    \frac{d(f(a))}{da} = 0 + \frac{1}{(a+1)^2}
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  4. #4
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    Yet a third way! \frac{a}{a+ 1}= a(a+ 1)^{-1}. Use the "product rule", rather than the "quotient rule", together with the chain rule:
    \left(a(a+1)^{1}\right)'= (a)'(a+ 1)^{1}+ a((a+1)^{-1})'
    = (1)(a+ 1)^{-1}+ (a)(-1(a+1)^{-2}(1))

    You could stop there or you could write it as
    = \frac{1}{a+ 1}- \frac{a}{(a+1)^2}= \frac{a+ 1}{(a+1)^2}- \frac{a}{(a+1)^2}= \frac{1}{(a+1)^2}
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