Second Derivatives


In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.


x²/³ + y²/³ = 1

Verified step by step guidance

1. Start by differentiating both sides of the equation x^(2/3) + y^(2/3) = 1 with respect to x. Use implicit differentiation for the y term.
2. For the term x^(2/3), apply the power rule: differentiate to get (2/3)x^(-1/3).
3. For the term y^(2/3), apply the chain rule: differentiate to get (2/3)y^(-1/3) * (dy/dx).
4. Set the derivative of the left side equal to the derivative of the right side (which is 0), and solve for dy/dx.
5. To find the second derivative d²y/dx², differentiate the expression for dy/dx with respect to x, applying implicit differentiation again, and express the result in terms of x and y.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. It involves differentiating both sides of an equation with respect to a variable, typically x, while treating other variables as implicit functions of x. This method is essential for finding dy/dx when y is not isolated.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This rule is crucial when applying implicit differentiation, as it allows for the differentiation of terms involving y with respect to x.

Second Derivative

The second derivative, denoted as d²y/dx², represents the derivative of the first derivative, providing information about the curvature or concavity of a function. In the context of implicit differentiation, finding the second derivative involves differentiating the expression for dy/dx again with respect to x, often requiring the use of the chain rule and implicit differentiation techniques.