0.0.1 ↑ 84. Hausaufgabe
0.0.1.1 ↑ Analysis-Buch Seite 149, Aufgabe 5
Leite ab:
- a)
\mathrm{f}(x) = x + \ln x; \quad \mathrm{f}'(x) = 1 + \frac{1}{x};
- b)
\mathrm{f}(x) = x \ln x; \quad \mathrm{f}'(x) = x \frac{1}{x} + \ln x = 1 + \ln x;
- c)
\mathrm{f}(x) = \ln -x; \quad \mathrm{f}'(x) = \frac{1}{x};
- d)
\mathrm{f}(x) = -\ln 2x; \quad \mathrm{f}'(x) = -\frac{2}{2x} = -\frac{1}{x} = \left(-\ln x\right)';
- e)
\mathrm{f}(x) = \ln x^2 = 2 \ln x; \quad \mathrm{f}'(x) = \frac{1}{x^2} \cdot 2x = \frac{2}{x};
- f)
\mathrm{f}(x) = \left(\ln x\right)^2; \quad \mathrm{f}'(x) = 2 \ln x \cdot \frac{1}{x};
- g)
\mathrm{f}(x) = \ln \sqrt{x}; \quad \mathrm{f}'(x) = \frac{1}{\sqrt{x}} \frac{1}{2 \sqrt{x}} = \frac{1}{2 x};
- h)
\mathrm{f}(x) = \sqrt{\ln x}; \quad \mathrm{f}'(x) = \frac{1}{2 \sqrt{\ln x}} \frac{1}{x};
- i)
\mathrm{f}(x) = \ln \sin x; \quad \mathrm{f}'(x) = \frac{1}{\sin x} \cdot \cos x;
- j)
\mathrm{f}(x) = \sin \ln x; \quad \mathrm{f}'(x) = \cos \ln x \cdot \frac{1}{x};
- k)
\mathrm{f}(x) = \ln x^e; \quad \mathrm{f}'(x) = \frac{1}{x^e} \cdot e x^{e - 1};
- l)
\mathrm{f}(x) = \ln e^x = x; \quad \mathrm{f}'(x) = \frac{1}{e^x} \cdot e^x = 1;