13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
x = e^y; (2, ln 2)

Verified step by step guidance

1. Start by differentiating both sides of the equation with respect to x. The equation given is x = e^y.
2. Differentiate the left side: The derivative of x with respect to x is 1.
3. Differentiate the right side: Use the chain rule. The derivative of e^y with respect to y is e^y, and then multiply by dy/dx because of the chain rule. So, the derivative is e^y * (dy/dx).
4. Set the derivatives equal to each other: 1 = e^y * (dy/dx).
5. Solve for dy/dx by dividing both sides by e^y: dy/dx = 1/e^y. Substitute the point (2, ln 2) into the equation to find the specific value of dy/dx at that 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 when necessary. This method is particularly useful for equations that define y implicitly in terms of x.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential in implicit differentiation, where y is often a function of x.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where a is a constant and x is the variable. In the context of the given problem, the equation x = e^y involves the natural exponential function, where e is the base of natural logarithms. Understanding the properties of exponential functions, including their derivatives, is crucial for applying implicit differentiation effectively.