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.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as dy/dx, representing the rate of change of y with respect to x.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If y = u/v, where u and v are functions of x, the derivative is given by dy/dx = (v(du/dx) - u(dv/dx)) / v². This rule is essential when differentiating functions that involve division.

Evaluating Derivatives at a Point

Evaluating a derivative at a specific point involves substituting the x-value into the derivative function obtained from differentiation. For example, if you find dy/dx and need to evaluate it at x=2, you substitute 2 into the derivative expression to find the slope of the tangent line to the curve at that point.