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Question Number 19903 by j.masanja06@gmail.com last updated on 17/Aug/17

$$\mathrm{by}\mathrm{use}\mathrm{the}\mathrm{first}$$$$\mathrm{principle},\mathrm{find}$$$$\mathrm{dy}/\mathrm{dx}\mathrm{of}$$$$\mathrm{y}=\cos (\mathrm{x}-\frac{\mathrm{\Pi }}{8})$$

Answered by icyfalcon999 last updated on 18/Aug/17

$$\frac{\mathrm{dy}}{\mathrm{dx}}=\underset{\mathrm{h}\to 0}{\lim }\frac{\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{h}}$$$$=\underset{\mathrm{h}\to 0}{\lim }\frac{\cos (\mathrm{x}+\mathrm{h}-\frac{\pi }{8})-\cos (\mathrm{x}-\frac{\pi }{8})}{\mathrm{h}}$$$$=\underset{\mathrm{h}\to 0}{\lim }\frac{\cos (\mathrm{h}+(\mathrm{x}-\frac{\pi }{8}))-\cos (\mathrm{x}-\frac{\pi }{8})}{\mathrm{h}}$$$$=\underset{\mathrm{h}\to 0}{\lim }\frac{\cos \left(\mathrm{h}\right)\cos (\mathrm{x}-\frac{\pi }{8})-\sin \mathrm{h}\sin (\mathrm{x}-\frac{\pi }{8})-\cos (\mathrm{x}-\frac{\pi }{8})}{\mathrm{h}}$$$$=\underset{\mathrm{h}\to 0}{\lim }\frac{\cos \left(\mathrm{h}\right)\cos (\mathrm{x}-\frac{\pi }{8})-\cos (\mathrm{x}-\frac{\pi }{8})-\sin \mathrm{h}\sin (\mathrm{x}-\frac{\pi }{8})}{\mathrm{h}}$$$$=\underset{\mathrm{h}\to 0}{\lim }\frac{\cos (\mathrm{x}-\frac{\pi }{8})(\cos \mathrm{h}-1)-\sin \mathrm{h}\sin (\mathrm{x}-\frac{\pi }{8})}{\mathrm{h}}$$$$=\underset{\mathrm{h}\to 0}{\lim }(\frac{\cos (\mathrm{x}-\frac{\pi }{8})(\cos \mathrm{h}-1)}{\mathrm{h}}-\frac{\sin \mathrm{h}\sin (\mathrm{x}-\frac{\pi }{8})}{\mathrm{h}})$$$$=\underset{\mathrm{h}\to 0}{\lim }(\frac{\cos \mathrm{h}-1}{\mathrm{h}}\bullet \cos (\mathrm{x}-\frac{\pi }{8}))-\underset{\mathrm{h}\to 0}{\lim }(\frac{\sin \mathrm{h}}{\mathrm{h}}\bullet \sin (\mathrm{x}-\frac{\pi }{8}))$$$$=\underset{\mathrm{h}\to 0}{\lim }\frac{\cos \mathrm{h}-1}{\mathrm{h}}\bullet \underset{\mathrm{h}\to 0}{\lim cos}(\mathrm{x}-\frac{\pi }{8})-\underset{\mathrm{h}\to 0}{\lim }\left(\frac{\sin \mathrm{h}}{\mathrm{h}}\right)\bullet \underset{\mathrm{h}\to 0}{\lim sin}(\mathrm{x}-\frac{\pi }{8})$$$$=0\bullet \cos (\mathrm{x}-\frac{\pi }{8})-1\left(\sin (\mathrm{x}-\frac{\pi }{8})\right)$$$$=-\sin (\mathrm{x}-\frac{\pi }{8})$$

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