obliczyc pochodna z definicji
: 7 sty 2009, o 12:39
\(\displaystyle{ f(x)= sin\sqrt{x}}\)
\(\displaystyle{ x_{0}=4}\)
z gory dziekuje,
\(\displaystyle{ x_{0}=4}\)
z gory dziekuje,
\(\displaystyle{ f'(x_0) = \lim_{h \to 0}\frac{f(x_0+h)-f(x_0)}{h}}\)
Zatem u nas:
\(\displaystyle{ f'(4) = \lim_{h \to 0}\frac{\sin \sqrt{4+h}-\sin 2}{h} =
\lim_{h \to 0}\frac{2\sin \frac{ \sqrt{4+h}-2}{2} \cos \frac{\sqrt{4+h}+2}{2} }{h} = \\ =
2\cos 2 \lim_{h \to 0}\frac{\sin \frac{ \sqrt{4+h}-2}{2}}{h} =
2\cos 2 \lim_{h \to 0}\frac{\sin \frac{ \sqrt{4+h}-2}{2}}{\frac{ \sqrt{4+h}-2}{2}} \frac{\frac{ \sqrt{4+h}-2}{2}}{h} = \\ =
2\cos 2 \lim_{h \to 0} \frac{\frac{ \sqrt{4+h}-2}{2}}{h} =
2\cos 2 \lim_{h \to 0} \frac{h}{2h (\sqrt{4+h}+2)} = 2 \cos 2 \frac{1}{8} = \frac{\cos 2}{4}}\)