Question

If \(y = cos^{-1}(sinx)\), where \(\frac{π}{2} < x < π\), then \(\frac{dy}{dx} = ...\)

• A. \(-1\)
• B. \(0\)
• C. \(1\)
• D. \(2\)

Hint:

Check the sign of cos x in the second quadrant.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
\(\cos^{-1}(\cos u)=|u|\) when \(-\pi\le u\le\pi\), and equals \(u\) only for \(0\le u\le\pi\). So we first write \(\sin x\) as a cosine.

Step 2: Rewrite:
\(\sin x=\cos\left(\dfrac\pi2-x\right)\). For \(\dfrac\pi2<x<\pi\), the angle \(u=\dfrac\pi2-x\) lies in \(\left(-\dfrac\pi2,0\right)\), which is negative.

Step 3: Simplify y:
\(y=\cos^{-1}\cos u=-u=x-\dfrac\pi2\).

Step 4: Differentiate:
\(\dfrac{dy}{dx}=1\). Option (C). The value \(-1\) would come from wrongly using \(u\) itself.

Final Answer:
On this interval y = x - pi/2, so dy/dx = 1. \[ \boxed{1} \]