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3.1 Derivatives Great Sand Dunes National Monument, Colorado Photo by Vickie Kelly, 2003 Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

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is called the derivative of at. We write: The derivative of f at a is the limit of the difference quotient as h approaches 0… There are many ways to write and to verbalize derivatives …

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One alternative definition of the derivative of f with respect to x at the point x = a …(handy for radical functions or fractions) The derivative of f (with respect to x) at a is …

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There are many traditions for writing derivatives -- some from Newton, some from Leibniz, some from function notation -- with subtle differences in emphasis, but the same overall meaning!

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f prime of x or the derivative of f with respect to x y prime dee y dee xor the derivative of y with respect to x dee f dee xor the derivative of f with respect to x dee dee x of f of x or the derivative of f of x with respect to x orthe derivative of y

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dx does not mean d times x ! dy does not mean d times y ! does not mean times !

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does not mean !

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In the future, all will become clearer...

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The derivative of f is the slope along f. The one-sided derivative is defined at the end point (original f has a closed interval!)

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A function is differentiable if it has a derivative everywhere (at every point!) in its domain. At each end point on closed intervals, continuous functions have one-sided derivatives defined. Differentiable regions are continuous and locally linear when you zoom in on them.

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3.2 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

3.2 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003.

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