5–8. Calculate dy/dx using implicit differentiation.
sin y+2 = x

Verified step by step guidance

1. Start by understanding the equation: \( \sin y + 2 = x \). We need to find \( \frac{dy}{dx} \) using implicit differentiation.
2. Differentiate both sides of the equation with respect to \( x \). Remember that \( y \) is a function of \( x \), so when differentiating \( \sin y \), use the chain rule.
3. The derivative of \( \sin y \) with respect to \( x \) is \( \cos y \cdot \frac{dy}{dx} \). The derivative of the constant 2 is 0, and the derivative of \( x \) is 1.
4. Set up the equation from the differentiation: \( \cos y \cdot \frac{dy}{dx} = 1 \).
5. Solve for \( \frac{dy}{dx} \) by isolating it: \( \frac{dy}{dx} = \frac{1}{\cos y} \). This is the expression for the derivative using implicit differentiation.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function that is not explicitly solved for one variable in terms of another. Instead of isolating y, we differentiate both sides of the equation with respect to x, applying the chain rule when differentiating terms involving y. This method is particularly useful when dealing with equations where y cannot be easily expressed as a function 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 when differentiating terms involving y.

Trigonometric Derivatives

Trigonometric derivatives refer to the derivatives of trigonometric functions, which are essential in calculus. For example, the derivative of sin(y) with respect to x is cos(y) * (dy/dx) due to the chain rule. Understanding these derivatives is crucial when differentiating equations that involve trigonometric functions, as they help in simplifying the expressions and finding the required derivatives.