The chain rule

In the last video step on derivatives, we will learn how to use the implicit differentiation to solve equations and how the chain rule relates.

• In most discussions of math, if the dependent variable (y) is a function of the independent variable (x), we express (y) in terms of (x). If this is the case, we say that (y) is an explicit function of (x). For example, when we write the equation (y=x^{2}+1), we are defining (y) explicitly in terms of (x).
• On the other hand, if the relationship between the function (y) and the variable (x) is expressed by an equation where (y) is not expressed entirely in terms of (x), we say that the equation defines (y) implicitly in terms of (x).
• Implicit differentiation allows us to find slopes of tangents to curves that are clearly not functions (they fail the vertical line test). We are using the idea that portions of (y) are functions that satisfy the given equation, but that (y) is not actually a function of (x).
• In general, an equation defines a function implicitly if the function satisfies that equation. An equation may define many different functions implicitly.