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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=(sinx)2(xlnx)sinx−xlnx(sinx).

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

(xlnx)=(x)lnx+x(lnx)=1lnx+xx1=lnx+1.

All in all, we thus obtain

sinxxlnx=(sinx)2(lnx+1)sinx−xlnxcosx=sinxlnx+1−sin2xxlnxcosx.