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

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

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

“f prime x”or “the derivative of f with respect to x” “y prime” “dee why dee ecks” or “the derivative of y with respect to x” “dee eff dee ecks” or “the derivative of f with respect to x” or “the derivative of f of x”

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

does not mean ! (except when it is convenient to think of it as division.) does not mean ! (except when it is convenient to think of it as division.)

The derivative is the slope function of the original function. The derivative is defined at the end points of a function on a closed interval. Given

GIVEN Find a formula for y’

Find the equation of the line tangent to at x = 4

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. 

To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical tangent discontinuity

Most of the functions we study in calculus will be differentiable.