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Aus Online Mathematik Brückenkurs 2

In this case, we have a slightly more complicated expression, but if we focus on the expression's outer form, we have essentially "something divided by sinx ". As a first step, we therefore use the quotient rule,

sinxxlnx=sinx2xlnxsinx−xlnxsinx 

We can, in turn, differentiate the expression xlnx by using the product rule:

xlnx=xlnx+xlnx=1lnx+xx1=lnx+1

All in all, we thus obtain

sinxxlnx=sinx2lnx+1sinx−xlnxcosx=sinxlnx+1−sin2xxlnxcosx