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8: More Derivatives

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”No one is original. Everyone is derivative.” - Sonny Rollins

Chapter eight will cut down the laborious task of taking a derivative into an efficient pattern recognition activity.

First we find the derivatives of constant and linear functions, then stumble into the pattern for the derivative of a power function. We also see how the derivative limit interacts with function addition, subtraction, multiplication, and division. The calculations for each shortcut are justified between statements, but the lazy reader only needs to memorize the bolded results.

We also find the derivatives of each of the trigonometric functions (after a little help from a new theorem called the Squeeze Theorem). These combined with the power functions give many different ways to practice the differentiation rules mentioned above.

We define the exponential function, including its domain and range and how to sketch it depending on the base of the function. The properties of exponential functions and taking their limits are also discussed.

We look into how our new exponential functions interact with the differentiation process to discover that one base in particular causes it to be its own derivative. This number is referred to as e, Euler’s number. We can once again get practice using our new exponential derivative shortcuts in the middle of more and more elaborate examples.


This page titled 8: More Derivatives is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Kenn Huber.

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