mtk: bukti turunan sin x adalah cos x

misalkan f(x)=sin x, maka

$f'(x)=\lim_{h\rightarrow 0}\frac{sin(x+h)-sinx}{h}$

$f'(x)=\lim_{h\rightarrow 0}\frac{sinxcosh+sinhcosx-sinx}{h}$

$f'(x)=\lim_{h\rightarrow 0}\frac{sinxcosh-sinx+sinhcosx}{h}$

$f'(x)=\lim_{h\rightarrow 0}\frac{sinx(cosx-1)+sinhcosx}{h}$

$f'(x)=\lim_{h\rightarrow 0}\frac{sinx(cosh-1)}{h}+\lim_{h\rightarrow 0}\frac{sinhcosx}{h}$

$f'(x)=sinx\lim_{h\rightarrow 0}\frac{cosh-1}{h}+cosx\lim_{h\rightarrow 0}\frac{sinh}{h}$

$f'(x)=(sinx)(0)+(cosx)(1)$

$f'(x)=cosx$

catatan:

$f'(x)=\lim_{h\rightarrow 0}\frac{cosh-1}{h}=\lim_{h\rightarrow 0}\frac{(cosh-1)(cosh+1)}{h(cosh+1)}=\lim_{h\rightarrow 0}\frac{cos^{2}h-1}{h(cosh+1)}$

$f'(x)=\lim_{h\rightarrow 0}\frac{-sin^{2}h}{h(cosh+1)}=-\lim_{h\rightarrow 0}\frac{sinh}{h}.\lim_{h\rightarrow 0}\frac{sinh}{cosh+1}=(-1)(0)=0$

mtk

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Sabtu, 01 Januari 2011

bukti turunan sin x adalah cos x

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