On-line Math 21

2.1  The definition of the derivative

2.1.1  Tangents and Velocity

What is the ``derivative''? The easy answer is that the derivative of a function f(x) at a point x is the slope of the curve y = f(x) at the point (x,f(x)) corresponding to that value of x . But, talking about the slope of a curve doesn't give any idea of the uses of the concept. Let's start with the idea of slope, however, since that is the easiest.

Example 1 Slope

Consider the function f(x) = x² . What do we mean by the slope of the curve y = x² at a point? Let's take the point with x -value 2, (2,4) . If the curve were a line, then its slope would obviously be the slope as is defined for a line, the rise over the run, or the change in y over the change in x ,

Dy Dx .

We often use the D to indicate the change of a quantity or variable.

But, this curve is not a line, and so we have to think a bit more subtly. We can easily think about the slope of the line between two points on the curve. This is called a secant line. I don't know why. Take one of the two points to be (2,4) , and the other to be nearby, say (2+h,(2+h)²) . The slope of the line connecting those points will be

slope = (2+h)2-4 (2+h)-2 = (4+4h+h2)-4 h = 4h+h2 h = 4+h.

So, in this case it's easy to see what the slope of the curve will be at (2,4) , you simply take h smaller and smaller, getting secant lines that are closer and closer to the line tangent to the curve at (2,4) . The limit as h® 0 gives the slope of the curve:

Slope of the curve  = lim h® 0  (2+h)2-4 (2+h)-2 = 4.

[Re-do the secant line animation here, but with the graph being f(x) = x^2]

Example 2 Velocity

High-school physics texts usually state that velocity = distance ¸ time, or some such. That's right as far as it goes, which really assumes that the speed (or velocity) is constant. If the speed isn't constant, which really is usually the case, the notion needs to be extended. First, let D₁ and D₂ be the positions of some moving object at times t₁ and t₂ . The average velocity over that time interval is the change in position divided by the change in time,

vav = D2-D1 t2-t1 .

Just as before, we can find, in a given example, the limit of the average velocity as the time interval decreases. That is the instantaneous velocity:

v = lim t2® t1  D2-D1 t2-t1 .

[Re-do the car example here. Exactly the same]

2.1.2  Definition of the derivative

The derivative of a function f(x) at a particular value x = a has the following formal definition. All it is is a general statement of the idea of finding the slope of the curve at (a,f(a)) .

Definition 1 Let f be a function which is defined on at least some open interval that contains the number x . Then, the derivative of f at x is the limit

f¢(a): = lim h® 0  f(a+h)-f(a) h .

The function f is differentiable at a if this limit exists. If f is differentiable at all x in its domain, then f is a differentiable function.

We usually don't worry about just the slope at one value, but want a formula to find the slope at any x in the domain of f , so that slope, as as function of x , is written as

f¢(x): = lim h® 0  f(x+h)-f(x) h .

The fraction

f(x+h)-f(x) h ,

for any given h , is called a difference quotient for the function f .

Example 3 Find f¢(2) for the function f(x) = x³ .

Solution

Example 4 Find f¢(x) for the function f(x) = Öx .

Solution

Example 5 Find the derivative of the function f(x) = 2^x .

Solution

The derivative as a function

Definition 2 Let f be a function which is defined on at least some open interval that contains the number x . Then, the derivative of f at x is the limit

f¢(x): = lim h® 0  f(x+h)-f(x) h .

The function f is differentiable at x if this limit exists. If f is differentiable at all x in its domain, then f is a differentiable function.

More on derivatives

Since the derivative is the limit as the change in x goes to 0 of the ratio of the change in f over the range in x , that is:

f¢(x) = lim Dx® 0  Df Dx = lim Dx® 0  f(x+Dx)-f(x) Dx ,

we often denote f¢(x) as¹

f¢(x) = df dx .

There are some other common terminologies for the derivative. If y = f(x) describes a function, then all of these mean the same thing, the derivative of the function:

f¢(x) = df dx = d dx (f(x)) = D(f)(x) = Dx(f)(x) = fx(x) = dy dx = D(y).

Footnotes:

¹The df/dx notation was developed by G. Leibniz, who was one of the first people to develop calculus. The other principal inventor of calculus was Issac Newton, who used the f¢(x) notation. Leibniz' notation seems very ornate, appropriately German. The comparison would have been better had Newton been French, since the f¢(x) notation almost seems French. But no, he was English. Actually, it seems that Newton preferred to denote the derivative with a dot over the function, but this standard notation has been attributed to him.

The links in the paragraphs above are to biographies of the two inventors of calculus, including something of the dispute about who invented what first.

Copyright (c) 2000 by David L. Johnson.