=for timestamp
Mi Mai 31 17:00:18 CEST 2006

=head2 84. Hausaufgabe

=head3 Analysis-Buch Seite 149, Aufgabe 5

Leite ab:

=over

=item a)

M<\mathrm{f}(x) = x + \ln x;
\quad \mathrm{f}'(x) = 1 + \frac{1}{x};>

=item b)

M<\mathrm{f}(x) = x \ln x;
\quad \mathrm{f}'(x) = x \frac{1}{x} + \ln x = 1 + \ln x;>

=item c)

M<\mathrm{f}(x) = \ln -x;
\quad \mathrm{f}'(x) = \frac{1}{x};>

=item d)

M<\mathrm{f}(x) = -\ln 2x;
\quad \mathrm{f}'(x) = -\frac{2}{2x} = -\frac{1}{x} = \left(-\ln x\right)';>

=item e)

M<\mathrm{f}(x) = \ln x^2 = 2 \ln x;
\quad \mathrm{f}'(x) = \frac{1}{x^2} \cdot 2x = \frac{2}{x};>

=item f)

M<\mathrm{f}(x) = \left(\ln x\right)^2;
\quad \mathrm{f}'(x) = 2 \ln x \cdot \frac{1}{x};>

=item g)

M<\mathrm{f}(x) = \ln \sqrt{x};
\quad \mathrm{f}'(x) = \frac{1}{\sqrt{x}} \frac{1}{2 \sqrt{x}} = \frac{1}{2
x};>

=item h)

M<\mathrm{f}(x) = \sqrt{\ln x};
\quad \mathrm{f}'(x) = \frac{1}{2 \sqrt{\ln x}} \frac{1}{x};>

=item i)

M<\mathrm{f}(x) = \ln \sin x;
\quad \mathrm{f}'(x) = \frac{1}{\sin x} \cdot \cos x;>

=item j)

M<\mathrm{f}(x) = \sin \ln x;
\quad \mathrm{f}'(x) = \cos \ln x \cdot \frac{1}{x};>

=item k)

M<\mathrm{f}(x) = \ln x^e;
\quad \mathrm{f}'(x) = \frac{1}{x^e} \cdot e x^{e - 1};>

=item l)

M<\mathrm{f}(x) = \ln e^x = x;
\quad \mathrm{f}'(x) = \frac{1}{e^x} \cdot e^x = 1;>

=back