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find derivative of log(cosx) w.r.t. 'x' using 1st principle

$Let f (x) = \log (\cos x) f (x + h) = \log [\cos (x + h)] f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} \Rightarrow f' (x) = \lim_{h \to 0} \frac{\log [\cos (x + h)] - \log (\cos x)}{h} \Rightarrow f' (x) = \lim_{h \to 0} \frac{1}{h} \log [\frac{\cos (x + h)}{\cos x}] \Rightarrow f' (x) = \lim_{h \to 0} \frac{1}{h} (\frac{\cos x . \cos h - \sin x . \sin h}{\cos x}) \Rightarrow f' (x) = \lim_{h \to 0} \frac{1}{h} (\cos h) - \lim_{h \to 0} \frac{(\sin h)}{h} \tan x \Rightarrow f' (x) = 0 - (\tan x) \lim_{h \to 0} \frac{\sin h}{h} \Rightarrow f' (x) = - \tan x$

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