frac{d}{dx} log tan left( frac{pi}{4} + frac{ | vedclass

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(C) Let $y = \log 	an \left( \frac{\pi}{4} + \frac{x}{2} ight)$.Applying the chain rule,we differentiate with respect to $x$:$\frac{dy}{dx} = \frac

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(C) Let $y = \log 	an \left( \frac{\pi}{4} + \frac{x}{2} ight)$.Applying the chain rule,we differentiate with respect to $x$:$\frac{dy}{dx} = \frac{1}{	an \left( \frac{\pi}{4} + \frac{x}{2} ight)} \cdot \sec^2 \left( \frac{\pi}{4} + \frac{x}{2} ight) \cdot \frac{d}{dx} \left( \frac{\pi}{4} + \frac{x}{2} ight)$$\frac{dy}{dx} = \frac{1}{	an \left( \frac{\pi}{4} + \frac{x}{2} ight)} \cdot \sec^2 \left( \frac{\pi}{4} + \frac{x}{2} ight) \cdot \frac{1}{2}$Using $	an 	heta = \frac{\sin 	heta}{\cos 	heta}$ and $\sec 	heta = \frac{1}{\cos 	heta}$:$\frac{dy}{dx} = \frac{\cos \left( \frac{\pi}{4} + \frac{x}{2} ight)}{\sin \left( \frac{\pi}{4} + \frac{x}{2} ight)} \cdot \frac{1}{\cos^2 \left( \frac{\pi}{4} + \frac{x}{2} ight)} \cdot \frac{1}{2}$$\frac{dy}{dx} = \frac{1}{2 \sin \left( \frac{\pi}{4} + \frac{x}{2} ight) \cos \left( \frac{\pi}{4} + \frac{x}{2} ight)}$Using the identity $\sin(2	heta) = 2 \sin 	heta \cos 	heta$:$\frac{dy}{dx} = \frac{1}{\sin \left( 2 \left( \frac{\pi}{4} + \frac{x}{2} ight) ight)} = \frac{1}{\sin \left( \frac{\pi}{2} + x ight)}$Since $\sin \left( \frac{\pi}{2} + x ight) = \cos x$:$\frac{dy}{dx} = \frac{1}{\cos x} = \sec x$.

$\frac{d}{dx} \log \tan \left( \frac{\pi}{4} + \frac{x}{2} \right) = $

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