Find the derivative of the function \( y=\cos ^{-1}(2 x+1) \) a) \( y^{\prime}=\frac{2}{\sqrt{1-2 x}} \) b) \( y^{\prime}=\frac{-2}{\sqrt{1-(2 x+1)^{2}}} \) c) \( y^{\prime}=\frac{4}{\sqrt{1-2 x}} \) d) \( y^{\prime}=\frac{1}{\sqrt{1-(2 x+1)^{2}}} \)

Let \( u = 2x + 1 \) so that \( y = \cos^{-1}(u) \). First, differentiate using the chain rule: \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \] The derivative of \( \cos^{-1}(u) \) with respect to \( u \) is: \[ \frac{dy}{du} = -\frac{1}{\sqrt{1 - u^2}} \] Since \( u = 2x + 1 \), we have: \[ \frac{du}{dx} = 2 \] Thus, \[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - u^2}} \cdot 2 = -\frac{2}{\sqrt{1 - (2x+1)^2}} \] So, the correct answer is: \[ \boxed{\frac{dy}{dx} = -\frac{2}{\sqrt{1 - (2x+1)^2}}} \] This corresponds to option (b).