Question

Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

`x/(1 + tan x)`

Answer 1

Let f(x) = `x/(1 + tan x)`

f'(x) = `((1 + tan x) d/dx(x) - x d/dx (1 + tan x))/(1 + tan x^2)`

f'(x) = `((1 + tan x) -x.d/dx (1 + tan x))/(1 + tan x^2)`    ...(i)

Let g(x) =1 + tan x. Accordingly: g (x + h) = 1 + tan (x + h).

By first principle,

g'(x) = `lim_(h->0) (g(x + h) -g(x))/h`

 = `lim_(h->0) [(1 + tan (x + h) -1 - tan x)/h]`

= `lim_(h->0)1/h [(sin (x + h))/cos (x + h) - (sin x)/(cos x)]`

= `lim_(h->0)1/h [(sin (x + h)cos x - sin x cos (x + h))/(cos (x + h) cos x)]`

= `lim_(h->0)1/h [sin (x + h - x)/(cos (x + h) cos x)]`

= `lim_(h->0)1/h [sin h)/(cos (x + h) cos x)]`

= `(lim_(h->0) sin h/h) . (lim_(h->0) 1/(cos (x + h) cos x))`

= `1 xx 1/(cos^2 x) = sex^2 x`

= `d/dx (1 + tan x) = sec^2 x`     ...(ii)

From (i) and (ii), we obtain

f'(x) = `(1 + tan x - x sec^2 x)/(1 + tan x)^2`