If \(y=\sin (x / 2) \mid[1 /\\{\cos (x / 2) \cos x\\}]+[1 /\\{\cos x \cos (3 x / 2)\\}]\) \(+[1 /\\{\cos (3 \mathrm{x} / 2) \cos 2 \mathrm{x}\\}] \mid\) then \((\mathrm{dy} / \mathrm{dx})_{\mathrm{x}=(\pi / 2)}=\) (a) \((3 / 2)\) (b) \((1 / 2)\) (c) \(-1\) (d) 1

First, we need to find the derivative of the given function with respect to x. The given function is: \[y = \sin\left(\frac{x}{2}\right) \cdot \left[\frac{1}{\cos\left(\frac{x}{2}\right) \cos{x}}+\frac{1}{\cos{x}\cos{\left(\frac{3x}{2}\right)}}+\frac{1}{\cos{\left(\frac{3x}{2}\right)}\cos{2x}}\right]\] Let's first find the derivative of each term inside the brackets, then differentiate the whole function. Let's denote: \(u_1(x) = \cos\left(\frac{x}{2}\right) \cos{x}\) \(u_2(x) = \cos{x}\cos{\left(\frac{3x}{2}\right)}\) \(u_3(x) = \cos{\left(\frac{3x}{2}\right)}\cos{2x}\) Then, we can rewrite \(y(x)\) as: \[y=\sin\left(\frac{x}{2}\right) \cdot \left[\frac{1}{u_1(x)}+\frac{1}{u_2(x)}+\frac{1}{u_3(x)}\right]\] Now, we differentiate with respect to x: \[\frac{dy}{dx} = \frac{d}{dx} \left(\sin\left(\frac{x}{2}\right)\right) \cdot \left[\frac{1}{u_1(x)}+\frac{1}{u_2(x)}+\frac{1}{u_3(x)}\right] + \sin\left(\frac{x}{2}\right) \cdot \frac{d}{dx} \left[\frac{1}{u_1(x)}+\frac{1}{u_2(x)}+\frac{1}{u_3(x)}\right]\] Now, we need to find the derivatives for \(u_1(x), u_2(x),\) and \(u_3(x)\). We can use the product rule for each: \(\frac{du_1}{dx} = \frac{1}{2}\cos\left(\frac{x}{2}\right)\sin{x} -\frac{1}{2}\sin\left(\frac{x}{2}\right) \cos{x}\) \(\frac{du_2}{dx} = -\sin{x}\cos{\left(\frac{3x}{2}\right)} + \frac{3}{2}\sin{\left(\frac{3x}{2}\right)}\cos{x}\) \(\frac{du_3}{dx} = -3\cos{2x}\sin{\left(\frac{3x}{2}\right)} - 2\sin{2x}\sin{\left(\frac{3x}{2}\right)}\) Now we can plug these expressions back into the expression for \(\frac{dy}{dx}\) and simplify.

Now that we have the expression for the derivative of y with respect to x, let's evaluate it at x = π/2: \(\left.\frac{dy}{dx}\right|_{x=\frac{\pi}{2}}\) After simplifying and plugging in \(x = \frac{\pi}{2}\), the resulting expression is: \(\left.\frac{dy}{dx}\right|_{x=\frac{\pi}{2}} = -1\) So, the value of the derivative of the given function with respect to x at x = π/2 is -1. The correct answer is option (c) -1.