Use implicit differentiation to find dy/dx.
cos y² + x = e^y

Verified step by step guidance

1. Start by differentiating both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating terms involving y, use implicit differentiation.
2. Differentiate the left side: For the term cos(y^2), use the chain rule. The derivative of cos(u) with respect to u is -sin(u), and the derivative of y^2 with respect to y is 2y. Therefore, the derivative of cos(y^2) with respect to x is -sin(y^2) * 2y * (dy/dx).
3. Differentiate the x term: The derivative of x with respect to x is simply 1.
4. Differentiate the right side: The derivative of e^y with respect to x is e^y * (dy/dx), using the chain rule.
5. Combine all the differentiated terms: Set the derivative of the left side equal to the derivative of the right side, resulting in the equation -sin(y^2) * 2y * (dy/dx) + 1 = e^y * (dy/dx). Solve this equation for dy/dx.

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 equations involving functions that are not easily isolated.

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 we often encounter nested functions.

Exponential and Trigonometric Functions

Understanding the properties of exponential and trigonometric functions is crucial for implicit differentiation. In the given equation, e^y represents an exponential function, while cos(y^2) involves a trigonometric function. Knowing how to differentiate these functions, including their derivatives (e^y for e^y and -sin(y^2) * 2y for cos(y^2)), is necessary to apply implicit differentiation correctly and find dy/dx.