«11C» 36. Hausaufgabe «PDF», «POD»

0.0.1 ↑ 36. Hausaufgabe

0.0.1.1 ↑ Selbstgestellte Aufgaben

1. f(x) = \sqrt{5x}; \Rightarrow f'(x) = \frac{\sqrt{5}}{2\sqrt{x}} = \frac{\sqrt{5x}}{2x};$f\left(x\right)=\sqrt{5x};\Rightarrow f^{\prime }\left(x\right)=\frac{\sqrt{5}}{2\sqrt{x}}=\frac{\sqrt{5x}}{2x};$

2. f(x) = x \cdot \left(2x^2 + 3\right) = 2x^3 + 3x; \Rightarrow f'(x) = 6x^2 + 3;$f\left(x\right)=x\cdot \left(2x^2+3\right)=2x^3+3x;\Rightarrow f^{\prime }\left(x\right)=6x^2+3;$

3. f(x) = \dfrac{\sqrt{x}}{x} = \dfrac{x^{\frac{1}{2}}}{x^1} = x^{-\frac{1}{2}}; \Rightarrow f'(x) = -\frac{1}{2}x^{-\frac{3}{2}} = -\frac{1}{2}\frac{1}{\sqrt{x^3}} = -\frac{1}{2}\frac{\sqrt{x^3}}{x^3};$f\left(x\right)=\frac{\sqrt{x}}{x}=\frac{x^{\frac{1}{2}}}{x^1}=x^{-\frac{1}{2}};\Rightarrow f^{\prime }\left(x\right)=-\frac{1}{2}{x}^{-\frac{3}{2}}=-\frac{1}{2}\frac{1}{\sqrt{x^3}}=-\frac{1}{2}\frac{\sqrt{x^3}}{x^3};$

4. f(x) = x^m \cdot x^n = x^{m + n}; \Rightarrow f'(x) = \left(m + n\right)x^{m + n - 1};$f\left(x\right)=x^m\cdot x^n=x^{m+n};\Rightarrow f^{\prime }\left(x\right)=\left(m+n\right)x^{m+n-1};$