The Derivative

Suppose that we wish to find the tangent to a given curve at a given point of that curve. Before we can attempt to do so, we must know what it means for a line to be tangent to a curve. It certainly means more than that the line intersects the curve only at one point, for this condition is satisfied in the following diagram by the solid black line:

Perhaps one might suggest that the slope of the line is concerned. (If $A$ is a vector parallel to the line, $A=(a_1,a_2)$ the ratio $\frac{a_2}{a_1}$ is the slope of the line if $a_1\ne 0$; the slope is undefined if $a_1=0$. Thus

$$\text{slope}=\frac{\text{vertical rise}}{\text{horizontal run}},$$

if defined.) We could then say the tangent at $x=1$ to the curve $y=x^2$ is the line through the point $(1,1)$ and having the same slope as the curve at that point. But now we have substituted the undefined notion of "slope of a curve" for the idea of "tangent to a curve."

Just as we have an intuitive notion which enables us to sketch in the diagram above a red line which approximates the tangent to the curve at a point, we have a notion of slope of the curve at a point. In fact, our intuitive notion of slope is probably just "the slope of the tangent line". We see from this example that the slope at different points, in this sense, may be different; it depends on the point, which in the case of each individual curve, depends on the x-coordinate of the point. Thus the slope is a function of $x$, i.e., there is a rule (perhaps quite complicated) by which we can assign to each number $x$ of a certain collection (not necessarily all $x$) a single number $s(x)$, which is in this case the slope of the curve at the given value of $x$.

But the question, "What is the slope?" remains unanswered, and the reason is a very simple one. For within the language of our algebra and geometry there is no combination of concepts capable of expressing what we intuitively feel that "slope" means. Therefore we must define the slope in terms which involve certain new notions.

Now suppose I want to find the slope of a mountainside at some point. What would I do? I could get an estimate of the slope by standing at the point of interest and sending a helper up the mountain a little way. Then we could compare the difference in height with the horizontal distance between us, giving a ratio which would serve as an approximate slope.

This would be the slope of the broken line in the diagram. It is evident from the diagram that if I were to send my helper very far away, perhaps over the top of the mountain, the approximate values I would get for the slope would probably be meaningless. On the other hand, if he came closer to me, the estimates we would get would be closer to what we would like to call the slope. More estimates could also be obtained by sending him down the mountain, and these would likewise be better if he were not sent so far.

Now let us try this procedure on the graph of the function $y=x^2$ at the point where $x=1$, i.e., at $(1,1)$. I stand at $(1,1)$ and send my helper up to where his x-coordinate is $1+h$. Then he is at the point $(1+h,(1+h{)}^2)$, because he is on the curve. The approximate value we get for the slope is

$$\frac{(1+h{)}^2-1}{(1+h)-1}=\frac{2h+h^2}{h}=2+h.$$

As the distance $h$ that he goes ahead is made shorter, or as $h$ approaches zero, our approximation for the slope becomes closer and closer to 2. This is likewise the case if we send him down the curve, for this simply amounts to using a negative $h$ in the above process. For instance, when $h=.0001$, we get an approximate slope of 2.0001; when $h=-.000001$, an approximate slope of 1.999999. Then we define the slope of $𝒚\mathbf{=}{𝒙}^{\mathbf{2}}$ at $𝒙\mathbf{=}\mathbf{1}$ to be 2, which is the number approached by the approximations to the slope as h approaches zero.

We have thus reduced the undefined geometrical term "slope" to the property of numbers which tells us that the values of $2+h$ are very close to 2 if h is very small. What is important is that there is a definite number 2 which has this property.

Now the same thing will happen if we take any value of $x$ for the function $y=x^2$. Instead of standing at $(1,1)$ we stand at $(x,x^2)$. Our helper goes to $(x+h,(x+h{)}^2)$. The approximate value for the slope is

$$\frac{(x+h{)}^2-x^2}{h}=\frac{2xh+h^2}{h}=2x+h.$$

Now as h approaches zero, these values $2x+h$ approach the number $2x$. Hence the slope of the function $y=x^2$ at any point $x$ is $2x$.

We have thus reduced the undefined geometrical term "slope" to the undefined numerical term "approaches." At this stage we shall rest in the process of definition. A precise definition of what is meant by "approaches" will be given later in the course when experience has prepared you for it, but we shall see in examples that it is quite easy to determine the slopes of many common curves without entering deeply into the concepts of "limit" and "approaching." Where special properties of these concepts are used, they should be noted, and careful proofs will be supplied at a later time.

Now let $y=f(x)$ be a function of $x$, and let $x_0$ be a certain fixed value for $x$. Suppose that the approximate slopes, or as we say, the difference quotients,

$$\frac{f(x_0+h)-f(x_0)}{h}$$

approach a single number $b$ as $h$ approaches zero. Then we define the slope of the graph of $𝒚\mathbf{=}𝒇\mathbf{(}𝒙\mathbf{)}$ at ${𝒙}_{\mathbf{0}}$, or the derivative of $𝒇\mathbf{(}𝒙\mathbf{)}$ at ${𝒙}_{\mathbf{0}}$, to be the number b. If there is no single number $b$ approached by the difference quotients (for instance if they approach 1 for positive h's approaching zero and $-1$ for negative h's approaching zero), we say that the slope (or derivative) of $y=f(x)$ at $x=x_0$ is undefined. If it is defined, we denote it by