### The Alternative Notation dy/dx for the Derivative

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.

Exercise: Explain why e^x ~ 1 + x is the linear approximation to the exponential function at x=0, and check its accuracy for small positive and negative values of x.