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 Rule
Power Rule Implies Chain Rule and Foreshadowing the Chain Rule the same ideas.
Maria Gaetana Agnesi
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.
The Derivative I Guessing the derivatives from the definition
The Derivative II Using difference Quotient to graph and guess
The Derivative Rules I The Power Rule
The Derivative Rules II Another approach to the Product Rule from my friend Paul Foerster
The Derivative Rules III The Quotient Rule developed using the Power Rule, an approach first suggested by Maria Gaetana Agnesi (1718 – 1799) who was helping her brother learn the calculus.
Next week: The Chain Rule.
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 Quotients Expands on the post immediately above and shows some numerical and graphical approaches using calculators and Desmos.
Tangents and Slopes You can use this Desmos app now to preview some of the things that he tangent line can tell us about the graph of a function or save (or reuse) it for later when concentrating on graphs. Discuss slope in relation to increasing, decreasing, concavity, etc.
Stamp out Slope-intercept Form
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 differentiability. If you zoom-in of the graph of a function (at a point where we will soon say the function is differentiable), the graph eventually looks like a line: the graph appears to be straight, and its slope is the number we will call its derivative.
To do this we need to zoom-in numerically. Zooming-in numerically is accomplished by finding the slope of a secant line, a line that intersects the graph twice, and then finding the limit of that slope as the two points come closer together.
This week’s posts start with local linearity and tangent lines. They lead to the difference quotient and the equation of the tangent line.
Local Linearity II Working up to difference quotient. The next post explains this in more detail.
Tangent Lines approaching difference quotients on calculator by graphing tan line.
Next week: Difference Quotients.
Karl Weierstrass (1815 – 1897) was the mathematician who (finally) formalized the 
definition of continuity. In that definition was the definition of limit. So, which came first – continuity or limit? The ideas and situations that required continuity could only be formalized with the concept of limit. So, looking at functions that are and are not continuous helps us understand what limits are and why we need them.
In the ideal world I mentioned last week, students would have plenty of work with continuous and not continuous functions. The vocabulary and notation, if not the formal definitions, would be used as early as possible. Then when students got to calculus, they would know the ideas and be ready to formalize the.
Using the definition of continuity to show that a function is or is not continuous at a point is a common question of the AP exams.
Continuity The definition of continuity.
Continuity Should continuity come before limits?
From One Side or the Other One-sided limits and one-sided differentiability
How to Tell Your Asymptote from a Hole in the Graph From the technology series. Showing holes and asymptotes on a graphing calculator.
Fun with Continuity Defined everywhere and continuous nowhere. Continuous only at a single point.
Intermediate Weather Using the IVT
Right Answer – Wrong Question Continuity or continuity on its domain
In an ideal world, I would like to have all students study limits in their precalculus course and know all about them when they get to calculus. Certainly, this would be better than teaching how to calculate derivatives in precalculus (after all derivatives are calculus, not pre… ). Here are a few of my previous posts on limit.
Finding Limits How to … and the use of “infinity” vs “DNE”
Dominance Finding limits the easy way.
Deltas and Epsilons Not tested on the AP Exams; here’s why.
Asymptotes The graphical manifestation of limits at or equal to infinity.
Next Week: Continuity
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Today we continue to look at some previous posts that I hope will help you and your students throughout the year. We begin with some posts on graphing calculator use and then a few general things in three posts on beginning the year, followed by some mathematics I hope students know before they start studying the calculus.
Graphing Calculators
There are four things that students may, and are required to know how to do, for the AP Exams. But graphing calculators are not required just to answer a few questions on the exams. They are to encourage investigations and experimentation in all math classes. And not just graphing calculator use but all kinds of appropriate technology. So, don’t restrict yourself and your students to only those operations required on the exam. That said, here are previous posts on exam calculator use; as the year goes on there will be other posts on the use of graphing calculators and other technology in your class.
Starting School
A little late perhaps …
These posts discuss basic ideas that I always hoped students knew about mathematics before starting calculus
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Teaching Calculus 