The notation *f*' for the derivative of a function *f* actually
harks back to Newton, who used {\dot *f*} to represent the
derivative of *f* with respect to *t* for a function of time.
A competing notation was invented by Newton's rival Leibnitz.
If *y* is a function of *x*, Leibnitz represents the derivative by
*dy/dx*
instead of our *y*'.
This notation has advantages and disadvantages. It is first
important to understand that, when *y = f(x)* and we write
*dy/dx = f*'
that the left-hand side is one symbol, and cannot be interpreted
as the quotient of two numbers *dy* and *dx*.
In fact the way to understand it is this: suppose we're at some
specified value *x* of the variable. Then take
*dy/dx* at *x* to be
*dy/dx*=lim_{Delta *x*--> 0}(Delta *y*)/(Delta *x*)
where Delta * x* is a possible change in *x* and
Delta *y*= *f*(*x*+Delta *x*)-*f*(*x*)
is the corresponding change in *y*.
Roughly speaking, we read
Delta *x* as ``the change in *x*'' and
*Delta y* as ``the change in *y*.''
So our equation becomes
*dy/dx* = lim_{Delta *x*--> 0}(Delta *y*)/(Delta *x*}
=lim_{Delta *x*--> 0} (*f*(*x*+Delta *x*)
- *f*(*x*))/(Delta *x*),
which is the usual definition of *f*'(*x*).

So one advantage of the notation is that it *reminds* us of
the definition. The *d*'s are like what is left of the Delta's
after the limiting process (``The ghosts of departed
quantities'' is what Bishop George Berkeley (1685-1753) called
them in his critique of the Calculus.) The *dy/dx* notation
also reminds us of the units for
the derivative. If *y* is measured in miles and *x* is measured in
hours, then *dy/dx* comes out in miles per hour, or miles/hour.

We will see other advantages of this notation when we study
linear approximation, the notation for the integral, and
separable differential equations.

A disadvantage of the Leibnitz notation is that it is more
awkward to write the derivative as a function. The value of
*dy/dx* at *2* becomes
*dy/dx*_{*x*=2}.

### The Leibnitz Notation and Linear Approximation

It makes sense to uncouple *dy* and *dx* and write
*dy = f*'(*x*)*dx*
if we remember that we are dealing with the * limit* of a ratio,
and interpret the equation as meaning ``in the limit, as the change
in *x* goes to zero, the change in *y* is *f*'(*x*)
times the change
in *x*.'' An essentially equivalent statement is
``The change in *y*
is approximately *f*'(*x*) times the change in *x*, with the
approximation getting better and better as the change in *x* goes to
zero.'' In more condensed notation this reads
Delta *y* ~ *f*'(*x*)Delta *x*
(~ = ``is approximately equal to'')
or, expanding Delta *y* and regrouping,
*f*(*x*+Delta x*)* ~ *f*(*x*) +
*f*'(*x*)Delta *x*.
The right hand side is called the *linear approximation* to *f*
at *x*.
In fact, as Delta *x* varies, the points
(*x* + Delta *x*, *f*(*x* + Delta *x*)) move
along the graph of *f*, while the points (*x* + Delta *x*,
*f*(*x*) + *f*'(*x*) Delta *x*)
move along a line. This line has slope *f'(x)* and passes through
(*x*, *f*(*x*))
when Delta *x*=0, so it is
precisely the tangent line to the graph of *f*
at the point (*x*, *f*(*x*)).
When we use the linear approximation we are
reading off values from the tangent line, rather than from the graph
of the function itself.
As an example, consider the function *y* = *f*(*x*) =
sin *x* near *x* = 1.
Here *f*(1) = sin 1 = 0.8414709, and the derivative at 1 is
*f*'(1) = cos 1 = 0.5403023.

Delta x f(x+Delta x) f(x) + f'(x)Delta x
-1 0 .30116868
-.1 0.7833270 .7874408
-.01 0.8360260 0.8318876
-.001 0.8409302 0.8409307
-.0001 0.84141695 0.84141695
.0001 0.84152501 0.841525015
.001 0.84201087 0.8420113
.01 0.8468318 0.8468740
.1 0.8912073 0.8955012
1 0.9092974 1.3817733

*Exercise:* Make up a similar table for
*y* = *f*(*x*) = *x*^2, at *x* = 1.