Slopes and Tangent Lines


In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.


y = (x + 3)/(1 – x), x = −2

Verified step by step guidance

1. Step 1: Identify the function to differentiate. The function given is \( y = \frac{x + 3}{1 - x} \).
2. Step 2: Apply the quotient rule for differentiation. The quotient rule states that if you have a function \( y = \frac{u}{v} \), then its derivative \( y' \) is given by \( y' = \frac{u'v - uv'}{v^2} \), where \( u = x + 3 \) and \( v = 1 - x \).
3. Step 3: Differentiate the numerator and the denominator separately. For \( u = x + 3 \), the derivative \( u' = 1 \). For \( v = 1 - x \), the derivative \( v' = -1 \).
4. Step 4: Substitute \( u, u', v, \) and \( v' \) into the quotient rule formula: \( y' = \frac{(1)(1 - x) - (x + 3)(-1)}{(1 - x)^2} \). Simplify the expression to find \( y' \).
5. Step 5: Evaluate the derivative at \( x = -2 \) to find the slope of the tangent line. Substitute \( x = -2 \) into the simplified expression for \( y' \) and calculate the slope.

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input. For a function y = f(x), the derivative, denoted as f'(x) or dy/dx, provides the slope of the tangent line to the curve at any point x. Understanding differentiation is crucial for solving problems involving rates of change and slopes of curves.

Quotient Rule

The Quotient Rule is a method for differentiating functions that are expressed as a quotient of two other functions, u(x) and v(x). It states that the derivative of y = u(x)/v(x) is given by (v(x)u'(x) - u(x)v'(x))/(v(x))^2. This rule is essential when dealing with rational functions, like y = (x + 3)/(1 - x), to find their derivatives accurately.

Slope of the Tangent Line

The slope of the tangent line to a curve at a given point is the value of the derivative of the function at that point. It represents the instantaneous rate of change of the function and is crucial for understanding the behavior of the function at specific values of the independent variable. For the function y = (x + 3)/(1 - x) at x = -2, calculating the derivative and evaluating it at x = -2 gives the slope of the tangent line.