Lösung 1.2:3c

Aus Online Mathematik Brückenkurs 2

We can write the expression as

1x1−x2=x1−x2−1 ,

and then we see that we have "something raised to −1", which can be differentiated one step by using the chain rule:

ddxx1−x2−1=−1x1−x2x1−x2=−1x1−x22x1−x2=−1x21−x2x1−x2

The next step is to differentiate the product x1−x2  using the product rule:

=−1x21−x2x1−x2+x1−x2=−1x21−x211−x2+x1−x2

The expression 1−x2  is of the type "root of something", so we use the chain rule to differentiate,

=−1x21−x21−x2+x121−x21−x2=−1x21−x21−x2+x121−x2−2x=−1x21−x21−x2−x21−x2

We write the expression on the right over a common denominator:

=−1x21−x21−x21−x22−x2=−1x21−x21−x21−x2−x2=−1−2x2x21−x232

NOTE: When we make simplifications of the form 1−x22=1−x2 , we assume that both sides are well defined (i.e. in this case that x lies between −1 and 1 ).