Differentiate the following w, r, t . x . cosec - 1 1 cos 5 x

Let y = cosec - 1 1 cos 5 x ⇒ y = cosec - 1 sec 5 x ⇒ y = cosec - 1 cosec π 2 - 5 x ⇒ y = π 2 - 5 x Differentiating both sides w.r.t x we get d y d x = d d x π 2 - 5 x ⇒ d y d x = - 5 x log 5

Differentiate the following w, r, t . x . cosec - 1 1 cos 5 x

Let y = cosec - 1 1 cos 5 x ⇒ y = cosec - 1 sec 5 x ⇒ y = cosec - 1 cosec π 2 - 5 x ⇒ y = π 2 - 5 x Differentiating both sides w.r.t x we get d y d x = d d x π 2 - 5 x ⇒ d y d x = - 5 x log 5

Differentiate the following w, r, t . x . cos - 1 1 + cos x 2

Let y = cos - 1 1 + cos x 2 ⇒ y = cos - 1 cos 2 x 2 ⇒ y = cos - 1 cos x 2 ⇒ y = x 2 Differentiating both sides w.r.t x we get d y d x = d d x x 2 ⇒ d y d x = 1 2

Differentiate the following w, r, t . x . cos - 1 1 - cos x 2 2

Let y = cos - 1 1 - cos x 2 2 ⇒ y = cos - 1 sin 2 x 2 2 ⇒ y = cos - 1 sin x 2 2 ⇒ y = cos - 1 cos π 2 - x 2 2 ⇒ y = π 2 - x 2 2 Differentiating both sides w.r.t x we get d y d x = d d x π 2 - x 2 2 ⇒ d y d x = - 2 x 2 ⇒ d y d x = - x

Differentiate the following w, r, t . x . tan - 1 1 - tan x 2 1 + tan x 2

Let y = tan - 1 1 - tan x 2 1 + tan x 2 ⇒ y = tan - 1 tan π 4 - tan x 2 1 + tan π 4 tan x 2 ⇒ y = tan - 1 tan π 4 - x 2 ⇒ y = π 4 - x 2 Differentiating both sides w.r.t x we get d y d x = d d x π 4 - x 2 ⇒ d y d x = - 1 2

Differentiate the following w, r, t . x . cosec - 1 1 4 cos 3 2 x - 3 cos 2 x

Let y = cosec - 1 1 4 cos 3 2 x - 3 cos 2 x ⇒ y = cosec - 1 1 cos 3 × 2 x ⇒ y = cosec - 1 1 cos 6 x ⇒ y = cosec - 1 sec 6 x ⇒ y = cosec - 1 cosec π 2 - 6 x ⇒ y = π 2 - 6 x Differentiating both sides w.r.t x we get d y d x = d d x π 2 - 6 x ⇒ d y d x = - 6

Differentiate the following w, r, t . x . tan - 1 1 + cos x 3 sin x 3

Let y = tan - 1 1 + cos x 3 sin x 3 ⇒ y = tan - 1 2 cos 2 x 6 2 sin x 6 cos x 6 ⇒ y = tan - 1 cos x 6 sin x 6 ⇒ y = tan - 1 cot x 6 ⇒ y = tan - 1 tan π 2 - x 6 ⇒ y = π 2 - x 6 Differentiating both sides w.r.t x we get d y d x = d d x π 2 - x 6 ⇒ d y d x = - 1 6

Differentiate the following w, r, t . x . cot - 1 sin 3 x 1 + cos 3 x

Let y = cot - 1 sin 3 x 1 + cos 3 x ⇒ y = cot - 1 2 sin 3 x 2 cos 3 x 2 2 cos 2 3 x 2 ⇒ y = cot - 1 sin 3 x 2 cos 3 x 2 ⇒ y = cot - 1 tan 3 x 2 ⇒ y = cot - 1 cot π 2 - 3 x 2 ⇒ y = π 2 - 3 x 2 Differentiating both sides w.r.t x we get d y d x = d d x π 2 - 3 x 2 ⇒ d y d x = - 3 2

Differentiate the following w, r, t . x . tan - 1 cos 7 x 1 + sin 7 x

Let y = tan - 1 cos 7 x 1 + sin 7 x ⇒ y = tan - 1 sin π 2 - 7 x 1 + cos π 2 - 7 x ⇒ y = tan - 1 2 sin π 4 - 7 x 2 cos π 4 - 7 x 2 2 cos 2 π 4 - 7 x 2 = tan - 1 sin π 4 - 7 x 2 cos π 4 - 7 x 2 = tan - 1 tan π 4 - 7 x 2 ⇒ y = π 4 - 7 x 2 Differentiating both sides w.r.t x we get d y d x = d d x π 4 - 7 x 2 ⇒ d y d x = - 7 2

Differentiate the following w, r, t . x . tan - 1 1 + cos x 1 - cos x

Let y = tan - 1 1 + cos x 1 - cos x ⇒ y = tan - 1 2 cos 2 x 2 2 sin 2 x 2 ⇒ y = tan - 1 cot 2 x 2 ⇒ y = tan - 1 cot x 2 ⇒ y = tan - 1 tan π 2 - x 2 ⇒ y = π 2 - x 2 Differentiating both sides w.r.t x we get d y d x = d d x π 2 - x 2 ⇒ d y d x = - 1 2