Math 205 JW Lecture 12 - The Chain Rule and Implicit Differentiation

Welcome to lecture 12 of Math 205 JW - Elements of Calculus.

In tonight's lecture we look at the chain rule and implicit differentiation.

The chain rule allows us to differentiate composite functions (functions of functions), i.e. to be able to differentiate something like (f(x))^n as opposed to simply x^n. We state the rule, illustrate its importance and do some examples.

We then move on to implicit differentiation. This allows us to find the derivative of a function that is defined implicitly, i.e. the x's and y's are mixed up in the function definition and perhaps cannot be separated to write in the form y = f(x). Or, more generally, it is when the dependent variable is not explicitly on one side of an equation on its own, but is mixed up with the independent variable in the equation. The trick here is to use the chain rule to differentiate each term, and solve for the derivative algebraically. We do several examples of this.

We will move on to logarithmic differentiation in the next lecture.