फलन `sqrt(tan x)` का x के सापेक्ष प्रथम सिद्धांत से अवकल - गुणांक ज्ञात कीजिए ।

Explanation

मान लीजिए ` f(x) = sqrt(tan x) " तो " f( x+h) = sqrt(tan ( x +h))`
` :. d/dx sqrt(tan x) = lim _(h to 0) (sqrt(tan(x+h))-sqrt(tan x))/h`
` = lim _( h to 0 ) sqrt( tan ( x +h) - sqrt(tan x ))/hxx(sqrt(tan (x+h))+ sqrt( tan x))/(sqrt( tan ( x +h))+ sqrt( tanx))`
`= lim _( h to 0) ( tan ( x +h)-tanx)/(h[sqrt( tan ( x+h))+sqrt(tanx)])`
` = lim_( h to 0) ( sin ( x+h)cos x - sin x cos ( x+h))/(h[sqrt( tan ( x +h))-sqrt( tan x)]cos ( x+h)cos x)`
` = lim _( h to 0) (sin h)/(h[sqrt(tan ( x +h))-sqrt( tan x)]cos ( x +h) cos x )`
`= lim _( h to 0) 1/([sqrt(tan ( x+h))+ sqrt(tan x)])xxlim_( h to 0) ((sin 0)/h) . lim _( h to 0) 1/( cos ( x+h) . cos x)`
` = 1/(2sqrt(tan x ))xx1/( cos^(2)x)=(sec^(2)x)/(2 sqrt(tanx))`