Implicit Differentiation

Posted on September 25, 2018 by Lin McMullin

Often a relation (an expression in x and y), that has a graph but is not a function, needs to be analyzed. But the relation is not or cannot be solved for y. What to do? The answer is to use the technique of implicit differentiation. Assume there is a way to solve for y and differentiate using the Chain Rule. Whenever you get to the y,“differentiate” it by writing dy/dx. Then solve for dy/dx.

Here are several previous posts on this topic and how to go about using it.

Implicit Differentiation and Inverses

Implicit differentiation of parametric equations These are BC topics

A Vector’s Derivative These are BC topics

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The Chain Rule

Posted on September 18, 2018 by Lin McMullin

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 posts listed below are ways to introduce and 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.

The Chain Rule

Revised from 9-19-2017

Derivative Formulae

Posted on September 11, 2018 by Lin McMullin

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.

Why Radians?

The Derivative I Guessing the derivatives from the definition

The Derivative II Using difference quotients 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.

Revised from 9-12-2017

Difference Quotients

Posted on September 4, 2018 by Lin McMullin

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 or the Desmos graph

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.

At Just the Right Time

Stamp out Slope-intercept Form

Updated from a post of 9-5-2017