13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
x = e^y; (2, ln 2)

Verified step by step guidance

1. Start by recognizing that the equation given is in implicit form: \( x = e^y \). This means that \( y \) is not isolated on one side of the equation.
2. To find the derivative \( \frac{dy}{dx} \), apply implicit differentiation to both sides of the equation with respect to \( x \). Differentiate \( x \) to get 1, and differentiate \( e^y \) using the chain rule to get \( e^y \cdot \frac{dy}{dx} \).
3. Set up the equation from the differentiation: \( 1 = e^y \cdot \frac{dy}{dx} \).
4. Solve for \( \frac{dy}{dx} \) by isolating it on one side of the equation: \( \frac{dy}{dx} = \frac{1}{e^y} \).
5. Substitute the given point \((2, \ln 2)\) into the expression for \( \frac{dy}{dx} \). Since \( y = \ln 2 \), calculate \( e^{\ln 2} \) which simplifies to 2, and find the slope at this point.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for curves defined by equations that cannot be easily rearranged.

Slope of a Curve

The slope of a curve at a given point represents the rate of change of the function at that point, which is mathematically defined as the derivative of the function. For a curve defined implicitly, the slope can be found by evaluating the derivative obtained through implicit differentiation. The slope is often denoted as 'dy/dx' and indicates how steep the curve is at the specified coordinates.

Exponential and Natural Logarithm Functions

Exponential functions, such as e^y, and natural logarithm functions, like ln(x), are fundamental in calculus. The function e^y is the inverse of ln(y), and they exhibit unique properties, such as the derivative of e^y being e^y dy/dx. Understanding these functions is crucial for evaluating expressions and derivatives involving exponential growth or decay, especially when working with implicit relationships.