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Question: If f(x) = \(\sqrt{1 - \sin 2x}\), then f ´(x) equals –...

If f(x) = 1−sin⁡2x\sqrt{1 - \sin 2x}, then f ´(x) equals –

A

–(cos x + sinx), for x ∈(π/4, π/2)

B

(cos x + sinx), for x ∈(0, π/4)

C

–(cos x + sinx), for x ∈(0, π/4)

D

(cos x – sinx), for x ∈(π/4, π/2)

Answer

–(cos x + sinx), for x ∈(0, π/4)

Explanation

Solution

f(x) = 1−sin⁡2x\sqrt{1 - \sin 2x}

= (sin⁡x−cos⁡x)2\sqrt{(\sin x - \cos x)^{2}} = |sinx – cos x| < 0

If x ∈(0,π4)\left( 0,\frac{\pi}{4} \right) ⇒ f(x) = –(sin x – cosx)

⇒ f ′(x) = – cos x – sin x

= –(sinx + cosx) if x ∈ (0,π4)\left( 0,\frac{\pi}{4} \right)