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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