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Math Help - Derivative of more Products than 2?

  1. #1
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    Smile Derivative of more Products than 2?

    I'm not sure where to begin with finding the derivative of this function:

    f(x) = x * e^x * cscx

    I understand the product rule and how it applies to problems that have two functions, but how would one incorporate a third function?

    I've using the product rule on the first two, x & e^x, and then used that derivative and putting it in as one of the two functions in the product rule, along with cscx, but this seems like a sloppy/invalid method.

    Thank you!
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  2. #2
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    I would do what you have suggested

     f(x) = x \times e^x \times \csc(x)

    Make u = x \times e^x and v = \csc(x)

    To find u' you will need to use the product rule again.
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  3. #3
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    Oh, well thank you! That's two you've answered for me.
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  4. #4
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    Think of (uvw)'= ((uv)w)' and treat (uv) as a single function:
    ((uv)w)'= (uv)'w+ (uv)w'. Now use the product rule on (uv). (uv)'= u'v+ uv" so (uvw)'= (u'v+ uv')w+ (uv)w'= u'vw+ uv'w+ uvw'.

    That extends very easily to any number of functions: (uvw...z)'= u'vw...z+ uv'w...z+ uvw'...z+ ...+ uvw...z'.

    Another way to get that is to use "logarithmic differentiation". If f= uvw...z then ln(f)= ln(uvw...z)= ln(u)+ ln(v)+ ln(w)+ ...+ ln(z). (ln(f))'= (1/f)f'= u'/u+ v'/v+ w'/w+ ... z'/z. Multiplying on both sides by f= uvw...z gives f'= u'vw...z+ uv'w...z+ uvw'...z+...+ uvw...z'
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