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Derivatives Sec. 3.1
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Mathematical Synonyms
Instantaneous Rate of Change of f(x) at x = a Slope of the tangent line of f(x) at x = a Derivative of f(x) at x = a Reminder: The slope of the secant line at x=a is the AVERAGE rate of change at a.
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There are many ways to write the derivative of
is called the derivative of at . We write: “The derivative of f with respect to x is …” There are many ways to write the derivative of
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“the derivative of f with respect to x”
“f prime x” or “y prime” “the derivative of y with respect to x” “dee why dee ecks” or “the derivative of f with respect to x” “dee eff dee ecks” or “the derivative of f of x” “dee dee ecks uv eff uv ecks” or
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Note: dx does not mean d times x ! dy does not mean d times y !
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Note: does not mean ! does not mean !
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Note: does not mean times !
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The derivative is the slope of the original function.
The derivative is defined at the end points of a function on a closed interval.
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y y' 41
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Let y=sin(x). Graph y’.
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A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. p
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HOMEWORK P #2, 4, 10, 11, 13-22, 25-28
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