If π ≤ x ≤ 2π and y = cos^–1 (cos x), find dy/dx.

Question

If π ≤ x ≤ 2π and y = cos^–1(cos x), find \(\cfrac{dy}{d\mathrm x} \).

Answer 1

y = cos^-1 (cos x)

for x (π, 2π)

y = cos^-1(cos x)

= cos^-1(cos (π + (x - π)))

= cos^-1(-cos (x - π))

= π – (x - π)

= 2π - x

[Since, cos(π + x) = - cos x and cos^-1(-x) = π - x]

So, \(\cfrac{dy}{d\mathrm x} \) = -1

For cos^-1(cos x), x = nπ are the ‘sharp corners’ where slope changes from 1 to -1 or vice versa, i.e., the points where the curves are not differentiable.