Question

निम्नलिखित फलन का अवकलज ज्ञात कीजिए।

sec x

Answer 1

माना f(x) = sec x

∴ f(x + h) = sec (x + h)

f(x + h) − f(x) = sec (x + h) − sec x

= `1/(cos(x + h)) - 1/(cos x)`

= `(cos x - cos (x + h))/(cos (x + h) cos x)`

= `(2sin (x + h/2)sin (h/2))/(cos (x + h) cos x`

`f'(x) = lim_(h → 0)(f(x + h) - f(x))/h` 

= `lim_(h → 0) (2 sin(x + h/2) sin (h/2))/(h cos (x + h) cos x)`

= `lim_(h → 0) (sin (x + h/2))/(cos (x + h)cos x). ((sin  h/2)/(h/2))`

= `(sin x)/(cos x. cos x)`         ......`[∵ lim_(h → 0)((sin  h/2)/(h/2)) = 1]`

= `1/(cos x). (sin x)/(cos x)`

= sec x tan x