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is the derivative of the function x(t). In other words, the derivative of a
function is a new function that tells how rapidly the original function was
changing. We now use neither Newton’s name for his technique (he called it
“the method of fluxions”) nor his notation. The more commonly used
notation is due to Newton’s German contemporary Leibnitz, whom the
English accused of plagiarizing the calculus from Newton. In the Leibnitz
notation, we write
v=dx
dt
to indicate that the function v(t) equals the slope of the tangent line of the
graph of x(t) at every time t. The Leibnitz notation is meant to evoke the
delta notation, but with a very small time interval. Because the dx and dt are
thought of as very small
.
x’s and
.
t’s, i.e. very small differences, the part of
calculus that has to do with derivatives is called differential calculus.
Differential calculus consists of three things:
• The concept and definition of the derivative, which is covered in
this book, but which will be discussed more formally in your math
course.
• The Leibnitz notation described above, which you’ll need to get
more comfortable with in your math course.
• A set of rules for that allows you to find an equation for the
derivative of a given function. For instance, if you happened to
have a situation where the position of an object was given by the
equation x=2t
7
, you would be able to use those rules to find dx/
dt=14t
6
. This bag of tricks is covered in your math course.
Chapter 2Velocity and Relative Motion