I had a question from a reader recently asking about how to determine the units for derivatives and integrals. Derivatives: The units of the derivative are the units of dy divided by the units of dx, or the units of the dependent variable (f(x) or y) divided by the units of the independent variable (x).…
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…
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…
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…
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 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…
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…
