MATH 131
Final Exam
4. Find $\frac{d y}{d x}$ for the following functions:
(a) (5 points) $\sqrt{2^{x}+\sin x-2 x^{2}+2}$
(b) (5 points) $\frac{x^{2} \ln x}{x^{2}+2 x+1}$

Step 1 ：Given the function \(f_a = \sqrt{2^{x} - 2x^{2} + \sin(x) + 2}\), we need to find its derivative. We use the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. The outer function here is the square root function and the inner function is \(2^{x} + \sin(x) - 2x^{2} + 2\).

Step 2 ：Applying the chain rule, we get \(\frac{2^{x}\ln(2)/2 - 2x + \cos(x)/2}{\sqrt{2^{x} - 2x^{2} + \sin(x) + 2}}\).

Step 3 ：\(\boxed{\frac{2^{x}\ln(2)/2 - 2x + \cos(x)/2}{\sqrt{2^{x} - 2x^{2} + \sin(x) + 2}}}\) is the derivative of the function \(f_a\).

Step 4 ：Given the function \(f_b = \frac{x^{2}\ln(x)}{x^{2} + 2x + 1}\), we need to find its derivative. We use the quotient rule, which states that the derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the square of the denominator.

Step 5 ：Applying the quotient rule, we get \(\frac{x^{2}(-2x - 2)\ln(x)}{(x^{2} + 2x + 1)^{2}} + \frac{2x\ln(x)}{x^{2} + 2x + 1} + \frac{x}{x^{2} + 2x + 1}\).

Step 6 ：\(\boxed{\frac{x^{2}(-2x - 2)\ln(x)}{(x^{2} + 2x + 1)^{2}} + \frac{2x\ln(x)}{x^{2} + 2x + 1} + \frac{x}{x^{2} + 2x + 1}}\) is the derivative of the function \(f_b\).