Question

Differentiate cos^-1(4cos^3x-3cosx) w.r.t 'x'​

Answer 1

Answer:

let y=cos^-1(4cos³x-3cosx)

y=cos^-1cos3x

y=3x

dy/dx=3

Answer 2

Differentiate cos¯¹(4cos³x - 3cosx) with respect to x.

we see, cos(3x) = cos(2x + x)

= cos2x cosx - sin2x sinx

[ ∵ cos2x = cos²x - sin²x , sin2x = 2sinx cosx ]

= (cos²x - sin²x) cosx - (2sinx cosx )sinx

= (cos²x - 1 + cos²x)cosx - 2sin²x cosx

= 2cos³x - cosx - 2(1 - cos²x)cosx

= 2cos³x - cosx - 2cosx + 2cos³x

= 4cos³x - 3cosx

i.e., cos3x = 4cos³x - 3cosx

now, putting cos3x in place of 4cos³x - 3cosx

⇒cos¯¹(4cos³x - 3cosx) = cos¯¹(cos3x)

we know, cos¯¹(cosA) = A

so, cos¯¹(cos3x) = 3x

hence, y = cos¯¹(4cos³x - 3cosx) = 3x

differentiating y with respect to x,

dy/dx = d(3x)/dx = 3

Therefore the differentiation of cos¯¹(4cos³x - 3cosx) is 3.