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This educational piece explores different ways to express derivatives in calculus, highlighting key notations such as “the derivative of f with respect to x” and the various forms like “f prime x” and “y prime.” It discusses the importance of derivatives, treating them as slopes of functions and their definitions on closed intervals. The text clarifies common misconceptions about notation, emphasizing that dx does not signify multiplication and that differentiability requires continuity and smooth transitions within a function's domain.
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Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington Derivatives Great Sand Dunes National Monument, Colorado
We write: There are many ways to write the derivative of is called the derivative of at . “The derivative of f with respect to x is …”
“the derivative of f with respect to x” “f prime x” or “y prime” “the derivative of y with respect to x” or “dee why dee ecks” “the derivative of f with respect to x” or “dee eff dee ecks” “the derivative of f of x” “dee dee ecks uv eff uv ecks” or
Note: dx does not mean d times x ! dy does not mean d times y !
does not mean ! does not mean ! Note: (except when it is convenient to think of it as division.) (except when it is convenient to think of it as division.)
does not mean times ! Note: (except when it is convenient to treat it that way.)
The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.
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
