calculus 3.1

3.1 DerivativesGreat Sand Dunes National Monument, Colorado

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

0

limh

f a h f ah

is called the derivative of at .f a

We write: 0

limh

f x h f xf x

h

“The derivative of f with respect to x is …”

There are many ways to write the derivative of y f x

f x “f prime x” or “the derivative of f with respect to x”

y “y prime”

dydx

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

dfdx

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

d f xdx “dee dee ecks uv eff uv ecks” or “the derivative

of f of x”( of of )d dx f x

dx does not mean d times x !

dy does not mean d times y !

dydx does not mean !dy dx

(except when it is convenient to think of it as division.)

dfdx

does not mean !df dx

(except when it is convenient to think of it as division.)

(except when it is convenient to treat it that way.)

d f xdx

does not mean times !ddx

f x

In the future, all will become clear.

y f x

y f x

The derivative is the slope of the original function.

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

2 3y x

2 2

0

3 3limh

x h xy

h

2 2 2

0

2limh

x xh h xyh

2y x

0lim 2h

y x h

0

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.