91Ó°ÊÓ

To find the derivative of the function \(y = x - \ln(2e^x + 1 + \sqrt{e^{2x} + 4e^x + 1})\), we apply the derivative rules for each part. For the first term, the derivative of \(x\) is simply \(1\). For the second term, we will apply the chain rule to find the derivative of the natural logarithm. The chain rule states that \( (\frac{d}{dx}f(g(x))) = f'(g(x)) \cdot g'(x)\). In this case, our \(f(x) = \ln(x)\), and our \(g(x) = 2e^x + 1 + \sqrt{e^{2x} + 4e^x + 1}\). Finding the derivatives: \(f'(x) = \frac{1}{x}\) \(g'(x) = \frac{d(2e^x + 1 + \sqrt{e^{2x} + 4e^x + 1})}{dx}\) Now we need to find the derivative of the function under the square root. Using the chain rule again, the derivative of the square root function is \(\frac{1}{2\sqrt{e^{2x} + 4e^x + 1}}\), and the derivative of its contents is \(\frac{d(e^{2x} + 4e^x + 1)}{dx}\).The derivative of \(e^{2x}\) is \(2e^{2x}\) applying the chain rule, and the derivative of \(4e^x\) is \(4e^x\). Putting it all together, we get: \(g'(x) = \frac{2e^x + 4e^x}{2\sqrt{e^{2x} + 4e^x + 1}}\) Finally, using the chain rule for the whole function we obtain its derivative: \(y' = 1 - \frac{1}{2e^x + 1 + \sqrt{e^{2x} + 4e^x + 1}} \cdot \frac{2e^x + 4e^x}{2\sqrt{e^{2x} + 4e^x + 1}}\) Step 2: Simplify the derivative

To simplify the expression, we can cancel out the square root in the denominator: \(y' = 1 - \frac{2e^x + 4e^x}{2e^x + 1 + \sqrt{e^{2x} + 4e^x + 1}}\) Now, we add the terms in the numerators: \(y' = 1 - \frac{6e^x}{2e^x + 1 + \sqrt{e^{2x} + 4e^x + 1}}\) Step 3: Critical points

To find critical points of the function, we will find the values of \(x\) for which the derivative is equal to zero or undefined. In this case, we need to solve the equation \(y' = 0\): \(0 = 1 - \frac{6e^x}{2e^x + 1 + \sqrt{e^{2x} + 4e^x + 1}}\) Critical points are found through algebraic manipulations and sometimes by using numerical-based methods, depending on the simplicity of the equation. In this specific exercise, finding the exact critical points by algebraic methods is not easily achievable. We recommend using numerical methods such as the Newton-Raphson method or software like MATLAB, Maple, or WolframAlpha to find the critical point values.