In Exercises 19–22, find the slope of the curve at the point indicated.


y = (x − 1) / (x + 1), x = 0

Verified step by step guidance

1. To find the slope of the curve at a given point, we need to find the derivative of the function. The function given is \( y = \frac{x - 1}{x + 1} \).
2. Use the quotient rule to differentiate the function. The quotient rule states that if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \). Here, \( u = x - 1 \) and \( v = x + 1 \).
3. Differentiate \( u \) and \( v \): \( u' = 1 \) and \( v' = 1 \).
4. Substitute \( u, v, u', \) and \( v' \) into the quotient rule formula: \( y' = \frac{(1)(x + 1) - (x - 1)(1)}{(x + 1)^2} \).
5. Simplify the expression for \( y' \) and then substitute \( x = 0 \) into the derivative to find the slope of the curve at the point \( x = 0 \).

Key Concepts

Derivative

The derivative of a function at a point provides the slope of the tangent line to the curve at that point. It is a fundamental concept in calculus used to determine how a function changes as its input changes. For the function y = (x − 1) / (x + 1), finding the derivative will help us calculate the slope at x = 0.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the quotient of two differentiable functions. If y = u/v, where both u and v are functions of x, the derivative y' is given by (u'v - uv')/v². Applying the quotient rule to y = (x − 1) / (x + 1) will allow us to find its derivative.

Substitution

Substitution involves replacing a variable with a specific value to evaluate a function or its derivative at that point. After finding the derivative of y = (x − 1) / (x + 1), substituting x = 0 into the derivative will yield the slope of the curve at the specified point.