Differentiate `logsecx` from first principles.

Explanation

`f^{prime}(x)=lim _{h Rightarrow 0} frac{f(x+n)-f(x)}{h}`

`=lim _{h Rightarrow 0} frac{log sec (x+h)-log sec x}{h}`

`=lim _{h Rightarrow 0} frac{log cos x-log cos (x+h)}{h}`

`=lim _{h Rightarrow 0} log (frac{cos x}{cos (x+h)}) times frac{1}{h}`

`=lim _{h Rightarrow 0} frac{log [1+frac{cos x}{cos (x+h)}-1]}{h}`

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

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