§3.2 – The Derivative FunctionOctober 2, 2015

Definition of The Derivative Function

The function defined by the formula

is called the derivative of with respect to . The domain of consists of all in the domain of for which the limit exists

h

xfhxfxf

h

)()(lim)(

0

'

Notation: - all nouns meaning the derivative function

Read “ prime” - a verb with a noun meaning take the

derivative of the given function

Example: Find the derivative with respect to of Graph and together.

Normal Line

The normal line to a curve at a point is the line perpendicular to the tangent at that point.

Example: Write the equation for the normal line to the curve at .

Derivatives

Secant line of a graph of displacement () vs time will give the average velocity within an interval

Tangent line of a graph of displacement vs time will give the instantaneous velocity at a certain point

h

tfthfv

hins

)()(lim 00

0

Remember these are slopes of a tangent line of the curve Positive slope = positive velocity (forward) Negative slope = negative velocity (backwards) Zero slope = zero velocity (stopped) Maximum slope = maximum velocity Minimum slope = minimum velocity

The tangent line will give the instantaneous rate of change of anything

The secant line will give the average rate of change of anything

Ex: A penny is dropped from the empire state building by some idiot who wants to see if it will kill anybody. The position function of the penny is 1) Find its average velocity during its trip to the ground 2) Find its instantaneous velocity right before it hits the ground.