\((\mathrm{d} / \mathrm{d} \mathrm{x})\left[\log (1+\sin \mathrm{x})+\log (\sec \\{(\pi / 4)-(\mathrm{x} / 2)\\})^{2}\right]=\) (a) 0 (b) \(4 \mid[(\cos x-\tan x) /(\sin x+\cos x)]\) (c) \(\log _{\mathrm{e}} 2\) (d) \(-\log _{\mathrm{e}} 2\)

The given function is: \[f(x) = \log(1 + \sin x) + \log(\sec{(\frac{\pi}{4} - \frac{x}{2})}^{2})\] We'll apply the rules of differentiation for logarithmic functions: \(\frac{d}{dx}\log(u) = \frac{u'}{u}\) and for the trigonometric functions (sin, cos, sec) and the chain rule when needed.

Let's differentiate \(\log(1 + \sin x)\) with respect to x. Using the chain rule, we have: \[\frac{d}{dx}\log(1 + \sin x) = \frac{d}{dx}(1 + \sin x) \cdot \frac{1}{1+\sin x}\] Now, differentiate the inner function: \[\frac{d}{dx}(1 + \sin x) = \cos x\] So, the derivative of the first term is: \[\frac{d}{dx}\log(1 + \sin x) = \frac{\cos x}{1+\sin x}\]

Now, let's differentiate \(\log(\sec{(\frac{\pi}{4} - \frac{x}{2})}^{2})\) with respect to x. First, differentiate the outer function using the logarithmic rule: \[\frac{d}{dx}\log(\sec{(\frac{\pi}{4} - \frac{x}{2})}^{2}) = \frac{d}{dx}{(\sec{(\frac{\pi}{4} - \frac{x}{2})}^{2})} \cdot \frac{1}{\sec{(\frac{\pi}{4} - \frac{x}{2})}^{2}}\] Now, differentiate the inner function using the chain rule: \[\frac{d}{dx}{(\sec{(\frac{\pi}{4} - \frac{x}{2})}^{2})} = 2 \sec{(\frac{\pi}{4} - \frac{x}{2})} \cdot \sec{(\frac{\pi}{4} - \frac{x}{2})} \tan{(\frac{\pi}{4} - \frac{x}{2})} (-\frac{1}{2})\] Thus, the derivative of the second term is: \[\frac{d}{dx}\log(\sec{(\frac{\pi}{4} - \frac{x}{2})}^{2}) = -\sec{(\frac{\pi}{4} - \frac{x}{2})} \tan{(\frac{\pi}{4} - \frac{x}{2})}\]

Now, add the derivatives of the first and second terms: \[\frac{d}{dx}f(x) = \frac{\cos x}{1+\sin x} - \sec{(\frac{\pi}{4} - \frac{x}{2})} \tan{(\frac{\pi}{4} - \frac{x}{2})}\] Using the trigonometric identities, \(\sec{(\frac{\pi}{4} - \frac{x}{2})} = \cos{(\frac{x}{2})}\) and \(\tan{(\frac{\pi}{4} - \frac{x}{2})} = 1 - \tan{(\frac{x}{2})}\), we can rewrite the expression as: \[\frac{d}{dx}f(x) = \frac{\cos x}{1+\sin x} - \cos{(\frac{x}{2})}(1 - \tan{(\frac{x}{2})})\] Looking at the given options: (a) 0 (b) \(4 \mid[(\cos x-\tan x) /(\sin x+\cos x)]\) (c) \(\log _{\mathrm{e}} 2\) (d) \(-\log _{\mathrm{e}} 2\) It seems that none of the given options matches the derivative of the given function.