Implicit Differentiation and Inverses

Implicit differentiation of relations is done using the Chain Rule.  Implicit Differentiation (from last Friday’s post. I discovered I never did a post on this topic before!) Implicit differentiation of parametric equations A Vector’s Derivative The inverse series  This series of posts reviews and expands what students know from pre-calculus about inverses. This leads to…

Implicit Differentiation

I discovered in doing next week’s post that I apparently never wrote about implicit differentiation. So here goes – an extra post this week! Implicit Differentiation The technique of implicit differentiation allows you to find slopes of relations given by equations that are not written as functions, or may even be impossible to write as…

The Chain Rule

Most of the function students are faced with in beginning calculus are compositions of the Elementary Functions. The Chain Rule allows you to differentiate composite functions easily. The posted listed below are ways to introduce and then use the Chain Rule. Experimenting with a CAS – Chain Rule  Using a CAS to discover the Chain…

Differentiation Techniques

So, no one wants to do complicated limits to find derivatives. There are easier ways of course. There are a number of quick ways (rules, formulas) for finding derivatives of the Elementary Functions and their compositions. Here are some ways to introduce these rules; these are the subject of this week’s review of past posts. Why…

Difference Quotients

Difference quotients are the path to the definition of the derivative. Here are three posts exploring difference quotients. Difference Quotients I  The forward and backward difference quotients Difference Quotients II      The symmetric difference quotient and seeing the three difference quotients in action.  Showing that the three difference quotients converge to the same value. Seeing Difference…

Working up to the derivative.

While limit is what makes all of the calculus work, people usually think of calculus as starting with the derivative. The first problem in calculus is finding the slope of a line tangent to a graph at a point and then writing the equation of that tangent line. Local Linearity is the graphical manifestation of…

Other Topics (Type 7)

Any topic in the Course and Exam Description may be the subject of a free-response question. The two topics listed here have been the subject of full free-response questions or major parts of them. Implicitly defined relations and implicit differentiation These questions may ask students to find the first or second derivative of an implicitly…