13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
cos y = x; (0, π/2)

Verified step by step guidance

1. Start by differentiating both sides of the equation with respect to x. The equation is cos(y) = x.
2. Differentiate the left side: The derivative of cos(y) with respect to x is -sin(y) * (dy/dx) because of the chain rule.
3. Differentiate the right side: The derivative of x with respect to x is 1.
4. Set the derivatives equal to each other: -sin(y) * (dy/dx) = 1.
5. Solve for dy/dx by isolating it: dy/dx = -1/sin(y).

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 y in terms of x, we differentiate both sides of the equation with respect to x, treating y as a function of x. This method is particularly useful for equations that are difficult or impossible to rearrange.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When using implicit differentiation, the chain rule is applied to account for the derivative of y with respect to x, denoted as dy/dx. This means that when differentiating a function of y, we multiply by dy/dx to reflect the dependence of y on x.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are essential in calculus for modeling periodic phenomena and solving various problems. In the context of the given equation, cos(y) = x, understanding the properties and derivatives of these functions is crucial for finding dy/dx. The derivative of cos(y) involves the chain rule and is equal to -sin(y) * dy/dx.