66–71. Higher-order derivatives Find and simplify y''.


x + sin y = y

Verified step by step guidance

1. First, identify the given equation: \( x + \sin(y) = y \). We need to find the second derivative \( y'' \).
2. Differentiate both sides of the equation with respect to \( x \) to find the first derivative \( y' \). Remember to use the chain rule for \( \sin(y) \), which gives \( \cos(y) \cdot y' \).
3. The differentiation of the left side \( x + \sin(y) \) with respect to \( x \) results in \( 1 + \cos(y) \cdot y' \). The right side \( y \) differentiates to \( y' \). Set these equal: \( 1 + \cos(y) \cdot y' = y' \).
4. Solve for \( y' \) by isolating it on one side of the equation. This involves rearranging the terms to get \( 1 = y' - \cos(y) \cdot y' \), which simplifies to \( 1 = y'(1 - \cos(y)) \).
5. Now, differentiate \( y' = \frac{1}{1 - \cos(y)} \) with respect to \( x \) to find \( y'' \). Use the quotient rule and chain rule as needed, and simplify the expression to obtain \( y'' \).

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In this case, we have an equation involving both x and y, and we need to differentiate with respect to x while treating y as a function of x. This involves applying the chain rule to account for the derivatives of y.

Higher-Order Derivatives

Higher-order derivatives refer to the derivatives of a function beyond the first derivative. The second derivative, denoted as y'', provides information about the curvature of the function and can indicate concavity. To find y'', we first need to find the first derivative y' and then differentiate it again, applying implicit differentiation as necessary.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. When differentiating y with respect to x, we apply the chain rule to account for the fact that y is a function of x. This means that when we differentiate terms involving y, we must multiply by dy/dx, which represents the derivative of y with respect to x.