Evaluate dy/dx and dy/dx|x=2 if y= x+1/x+2

Verified step by step guidance

1. Step 1: Identify the function y = \frac{x+1}{x+2}. This is a rational function, which is a ratio of two polynomials.
2. Step 2: Use the quotient rule to find \frac{dy}{dx}. The quotient rule states that if y = \frac{u}{v}, then \frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}, where u = x+1 and v = x+2.
3. Step 3: Differentiate the numerator and the denominator separately. For u = x+1, \frac{du}{dx} = 1. For v = x+2, \frac{dv}{dx} = 1.
4. Step 4: Substitute the derivatives into the quotient rule formula: \frac{dy}{dx} = \frac{(x+2) \cdot 1 - (x+1) \cdot 1}{(x+2)^2}.
5. Step 5: Simplify the expression for \frac{dy}{dx} and then evaluate it at x = 2 by substituting x = 2 into the simplified expression.