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Lösung 1.2:3f

Aus Online Mathematik Brückenkurs 2

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We have no differentiation rule for a function raised to another function, but instead we use the formula

ab=elnab=eblna

which, in our case, gives

xtanx=etanxlnx. (*)

Now, we obtain the derivative by first using the chain rule

ddxetanxlnx=etanxlnxtanxlnx 

and then the product rule

=etanxlnx(tanx)lnx+tanx(lnx)=etanxlnx1cos2xlnx+tanxx1=etanxlnxlnxcos2x+xtanx=xtanxlnxcos2x+xtanx

where we have used (*) in reverse.

Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_1.2:3f“