Differentiate `e^(sqrt(tanx))` from the first principle.

Explanation

Let,`f(x)=e^(sqrttanx)`
=`lim_(h->0)(f(x+h)−f(x))/h`
=`lim_(h->0)(e^(sqrt(tan(x+h)))−e^(sqrt(tanx)))/h `

=`lim_(h->0)e^(sqrt(tan(x+h))){(e^(sqrt(tan(x+h))−tanx)−1)/h} `


=`e^(sqrt(tanx)).1.lim_(h->0)(tan(x+h)−tanx)/hxx (1/(sqrt(tan(x+h))+sqrt(tanx)) )`

=`(e^sqrt(tanx)).1.lim_(h->0)sinh/(hcosx(x+h)cosx)xx 1/(sqrt(tan(x+h))+sqrt(tanx))`
=(`e^(sqrt(tanx)))/(2sqrt(tanx)).(1/cos^2x) `

=`(sec^2x)/2.(e^tanx/(sqrt(tanx)))`