Derivatives calculus

Question

Find the derivative of the function

f(x) = cos^-1(sin 2x)

 

 

Thanks

 

Answer 1

Write cos^-1x = y or cos y = x.

Differentiate both sides of cos y = x to obtain

-sin y(dy/dx) = 1.

Divide both sides of -sin y(dy/dx) = 1 by -sin y

to yield dy/dx = (1/-sin y) or 1/[-√(1-cos²y)].

Replace y with cos^-1x and cos y with x in

dy/dx = 1/[-√(1-cos²y)] to obtain

d(cos^-1x)/dx = -1/[√(1-x²)]

As a general rule, d(cos^-1u)/dx = -(du/dx)/[√(1-u²)]

For f(x) = cos^-1(sin 2x), one then writes

d[cos^-1(sin 2x)]/dx = -2cos (2x)/√(1- sin²(2x))

or -2cos (2x)/√(cos²(2x)) or -2cos (2x)/cos(2x) or -2.