Differentiate the following w.r.t., x
(i) `tan^3 x` (ii) `(cos x)/((1 - sin x))` (iii) `sin (ax + b)^2`

Answer

(i) `3 tan ^2 x sec^2 x` (ii) `(1)/((1 - sin x))` (iii) `2a (ax + b) cos (ax +b)^2`

Explanation

(i)`y= tan^3x = (tan x)^3, (dy)/(dx) = (d)/(dx)(tanx)^3 = 3(tan x)^2(d)/(dx) (tan x) = 3 tan^2 x sec^2x`
(ii) `y= (cos x)/(1 -sin x) : (dy)/(dx) = ((1-sin s)(d)/(dx) cos x - cos x(d)/(dx) (1 - sinx))/((1 - sinx)^2) = (-sin x +1)/((1 - sinx)^2) = (1)/((1-sinx))`
(iii) `y = sin (ax +b)^2, (dy)/(dx) = (d)/(dx) [sin (ax +b)^2] = cos (ax +b)^2 (d)/(dx) (ax +b)^2`
= `cos (ax +b)^2 xx2(ax +b) (d)/(dx) (ax +b) = cos (ax +b)^2 xx2(ax +b)xxa`
`=2a(ax +b )cos (ax +b)^2`