Question

If \( y (\cos x)^{\sin x} = (\sin x)^{\sin x} \), then the value of \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \) is:

• A. \( 0 \)
• B. \( 1 \)
• C. \( \sqrt{2} \)
• D. \( \frac{\sqrt{3}}{2} \)

Hint:

When differentiating functions of the form \( f(x)^{g(x)} \), use logarithmic differentiation. - Remember that \( \ln \sin x - \ln \cos x = \ln \tan x \).

Answer 1

The Correct Option is C

Step 1: Take the logarithm
\[ \ln y = \ln \left( \frac{(\sin x)^{\sin x}}{(\cos x)^{\sin x}} \right). \] Using logarithm properties: \[ \ln y = \sin x \ln \sin x - \sin x \ln \cos x. \] Step 2: Differentiate both sides
Using the derivative rule: \[ \frac{1}{y} \frac{dy}{dx} = \cos x \ln \sin x + \sin x \frac{\cos x}{\sin x} - \cos x \ln \cos x - \sin x \frac{\sin x}{\cos x}. \] Simplifying: \[ \frac{1}{y} \frac{dy}{dx} = \cos x (\ln \sin x - \ln \cos x) + \sin x \left(\frac{\cos x}{\sin x} - \frac{\sin x}{\cos x} \right). \] Step 3: Evaluate at \( x = \frac{\pi}{4} \)
\[ \ln \sin \frac{\pi}{4} = \ln \cos \frac{\pi}{4} = \ln \frac{\sqrt{2}}{2}. \] \[ \cos \frac{\pi}{4} (\ln \sin \frac{\pi}{4} - \ln \cos \frac{\pi}{4}) + \sin \frac{\pi}{4} (0) = \sqrt{2} (0). \] Thus, \[ \frac{dy}{dx} = \sqrt{2}. \] So, the correct answer is \( \boxed{\sqrt{2}} \).