If-x-1-2-1-differentiate-cos-1-2x-1-x-2-

Question Number 44267 by rahul 19 last updated on 25/Sep/18

I f x ϵ (\frac{1}{\sqrt{2}}, 1), d i f f e r e n t i a t e \cos^{- 1} (2 x \sqrt{1 - x^{2}}) .

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Sep/18

x = c o s α d x = - s i n α d α

y = {c o s}^{- 1} (2 c o s α s i n α)

y = {c o s}^{- 1} (s i n 2 α)

n o w x ϵ (\frac{1}{\sqrt{2}}, 1) α ϵ (0, \frac{Π}{4}) s o 2 α ϵ (0, \frac{Π}{2})

w h e n x = \frac{1}{\sqrt{2}} α = \frac{Π}{4}

x = 1 α = 0

y = {c o s}^{- 1} {c o s (\frac{Π}{2} - 2 α)}

y = \frac{Π}{2} - 2 α

\frac{d y}{d α} = - 2 \frac{d y}{d x} = \frac{d y}{d α} \times \frac{d α}{d x} = - 2 \times \frac{- 1}{s i n α} = \frac{2}{\sqrt{1 - x^{2}}}

o r a p p r o a c h

l e t x = s i n α

d x = c o s α d α

y = {c o s}^{- 1} (2 s i n α c o s α)

y = {c o s}^{- 1} (s i n 2 α)

x ϵ (\frac{1}{\sqrt{2}}, 1) α ϵ (\frac{Π}{4}, \frac{Π}{2}) b u t 2 α ϵ (\frac{Π}{2}, Π)

l e t 2 α = \frac{Π}{2} + β 2 d α = d β

y = {c o s}^{-} {s i n (\frac{Π}{2} + β)}

y = {c o s}^{- 1} {c o s β) y = β

d y = d β

\frac{d y}{d x} = \frac{d y}{d β} \times \frac{d β}{d α} \times \frac{d α}{d x} = 1 \times 2 \times \frac{1}{c o s α} = \frac{2}{\sqrt{1 - x^{2}}}

p l s c h e c k

Commented by tanmay.chaudhury50@gmail.com last updated on 25/Sep/18

m o s t w e l c o m e \dots y o u a r e a g o o d b o w l e r \dots

Commented by rahul 19 last updated on 25/Sep/18

thanks sir ! ��