Instantaneous Velocity

FYI:Think of x as a crude difference in x's.

Think of dx as a very fine difference in x's.

Think of t as a crude difference in t's.

Think of dt as a very fine difference in t's.

Topic 2.1 ExtendedE – The method of slopes

FYI: Leibniz invented calculus about a decade after Newton, apparently without having seen much of Newton's work on the subject. Leibniz' notation is still used today. Newton's is obscure and ignored.

We have two equivalent definitions of instantaneous velocity, both of which are hard to use:

x

t

dx

dt

limit

t→0

=

v =

x(t + t) - x(t)

t

dx

dt

limit

t→0

v =

=

Before we show yet another way to find the derivative we introduce a new notation, courtesy of Gottfried Wilhelm Leibniz (1646-1716).

The four-step process of taking the derivative is outlined above in the second form:

The whole process will be represented with the new symbol dx/dt: