AP Calc BC: AB Review Part 3: Implicit & Logarithmic Differentiation

This video is for my AP Calculus BC students who need to briefly review Implicit Differentiation with another look. You'll want to be familiar with how to take a derivative of a function with a variable in the base as well as the exponent(wink wink, hint hint), so get clear on that now before we do some more applied problems from Related Rates, as well as to prepare for some of the new concepts you'll be taking a look at later in your Calc BC course. The idea of implicit differentiation is motivated by the chain rule, and is pretty simple to execute once you get the hang of it again; the technique of taking a logarithm on both sides first is what makes it possible to take the derivative since no clean rule exists when there is a variable both in the base and the exponent. Give yourself about 12 minutes to try this mix of problems: free-response questions about a normal line and 2 more straightforward derivative problems. When you're ready, un-pause the video and check out how it was done here. Some Things to Keep in Mind: Implicit Derivatives really are just like the chain rule; don't overthink it. You really only have to be careful/aware of what variable is being differentiated and the variable with respect to which you are differentiating(I know that the way this sentence reads might sound kind of insane, but I'm pretty sure that it is grammatically correct; we are just used to butchering the English language. . .so much for not overthinking it.) Related Rates use the concept of Implicit Derivatives extensively, so be clear with how to execute the derivative technique first before you try to apply it with other critical thinking. Chapters 0:00 Introduction 1:00 First Glance at the Problems 2:00 Technique 3:30 Quick First Example 4:30 Problem One: Normal Line Analysis 7:00 Plug in the Point to Find Tangent Slope 7:45 Problem One Part B: Intersection w/ Original Curve 9:30 Solving a Simple Quadratic Equation 10:45 Logarithmic Differentiation -- but Whyyy 11:20 Problem Two 12:45 Product Rule 13:15 Oversimplification? Mayhaps 15:30 Problem Three 16:30 Another Careful Product Rule 18:15 Some Concluding Blab