Compute the derivative of tan x. - Vidyadip

Question

Compute the derivative of tan x.

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Answer

Let f(x) = tan x.
Therefore,we have,
$\frac{d f(x)}{d x}=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$=\lim _{h \rightarrow 0} \frac{\tan (x+h)-\tan (x)}{h}$
$=\lim _{h \rightarrow 0} \frac{1}{h}\left[\frac{\sin (x+h)}{\cos (x+h)}-\frac{\sin x}{\cos x}\right]$
$=\lim _{h \rightarrow 0}\left[\frac{\sin (x+h) \cos x-\cos (x+h) \sin x}{h \cos (x+h) \cos x}\right]$
$\left.=\lim _{h \rightarrow 0} \frac{\sin (x+h-x)}{h \cos (x+h) \cos x} \text { (using formula for } \sin (\mathrm{A}+\mathrm{B})\right)$
$=\lim _{h \rightarrow 0} \frac{\sin h}{h} \cdot \lim _{h \rightarrow 0} \frac{1}{\cos (x+h) \cos x}$
$=1 \cdot \frac{1}{\cos ^{2} x}=\sec ^{2} x$

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