Differentiate each of the following from first principle: `t a n 2x`

Explanation

`frac{d}{d x}(f(x))=lim _{h rightarrow 0} frac{f(x+h)-f(x)}{h}`

`=lim _{h rightarrow 0} frac{tan (2 x+2 h)-tan (2 x)}{h}`

`=lim _{h rightarrow 0} frac{frac{sin (2 x+2 h)}{cos (2 x+2 h)}-frac{sin (2 x)}{cos (2 x)}}{h}`

`=lim _{h rightarrow 0} frac{sin (2 x+2 h) cos (2 x)-cos (2 x+2 h) sin (2 x)}{h cos (2 x+2 h) cos (2 x)}`

`=lim _{h rightarrow 0} frac{sin (2 x+2 h-2 x)}{h cos (2 x+2 h) cos (2 x)}`

`=frac{1}{cos 2 x} lim _{h rightarrow 0} frac{sin (2 h)}{2 h} times 2 times lim _{h rightarrow 0} frac{1}{cos (2 x+2 h)}`

`=frac{1}{cos 2 x} times 2 times frac{1}{cos 2 x}`

`=frac{2}{cos ^{2}(2 x)}`

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