Intuition for the Derivative at a Point

Sensitivity in the Real World

A derivative most essentially measures the "sensitivity" of one quantity to a small change in another quantity. Here are a few examples:

$\bullet$ How sensitive is a student's expected exam score to a small change in the amount of time the student spends studying?

$\bullet$ How sensitive is the amount of gas purchased at a certain gas station to a small change in the price of gas at that station?

$\bullet$ How sensitive is the location of a sprinter to a small change in the time elapsed since the beginning of the race? (You should realize that this basically asks for the velocity of the sprinter, which is why velocity is a derivative.)

These examples help show why derivatives are important; they factor into the student's decision of whether to keep studying and the gas station's decision of whether to raise or lower its price. As the third example shows, derivatives allow us to more precisely say what velocity really is.

Notice that to talk about the sensitivity of one quantity to a small change in another quantity, the first quantity must be a function of the second, that is, the first quantity must depend in a predictable way on the second.

For example, consider the first example above. If we're going to talk about the sensitivity, we need to be able to write the student's expected exam score (say, $s$) as a function of the amount of time (say, $t$) he/she spends studying. For any value of $t$, we should be able to plug it into the function and get a value of $s$. That is, there should be a function $f$ so that $s = f(t)$.

Here's another question, related to our second example above:

The point is this: to talk about sensitivity, we need a function, and in real-world applications, this often requires holding some quantities constant.

Sensitivity of Functions

Since sensitivity has to do with functions, let's think about some simple functions and their graphs. We are now moving away from real-world applications and moving toward abstract mathematical equations; we do this because the abstract equations are simpler to think about. However, keep the real-world applications in mind as we go!

Let's think about the function, $f(x) = x^{3} - 4x$. Here's its graph:

We'll be talking about the sensitivity of $y$ to a small change in $x$. Here's something we immediately notice:

The sensitivity can vary depending on where we start; in other words, the sensitivity can be different at different points of the function.

For now, let's start at $(0, 0)$.

In the questions above, we saw that the effects on $y$ of a small positive and negative change in $x$ are opposite. If you look at other points of this function, it seems like this is almost always the case for this function:

(The graph is reproduced below; see for yourself!)

$\bullet$ If a small increase in $x$ results in a sharp decrease in $y$, then a small decrease in $x$ results in a sharp increase in $y$. (e.g., $(0, 0)$)

$\bullet$ If a small increase in $x$ results in a sharp increase in $y$, then a small decrease in $x$ results in a sharp decrease in $y$. (e.g., $(2, 0)$)

Rather than talking about the sensitivity to each side of a point, it's more convenient to combine the two sides into one. To do this, we capitalize on the pattern that the effects of a small positive and negative change are often opposites:

$\bullet$ At $(2, 0)$, we say that the sensitivity of $f(x)$ is positive, by which we mean that a small increase in $x$ results in an increase in $y$, and a small decrease in $x$ results in a decrease in $y$.

$\bullet$ At $(0, 0)$, we say that the sensitivity of $f(x)$ is negative, by which we mean that a small increase in $x$ results in a decrease in $y$, and a small decrease in $x$ results in an increase in $y$.

We'd like to express the sensitivity as a number--this will allow to say not only whether $y$ increases or decreases as a result of a change in $x$, but how sharply $y$ increases or decreases.

As you may already realize, we can do this with the concept of slope!

This is tricky, since slope is only defined for lines, but here's how we can think of it: At any point of the function, zoom in until the function looks like a line. Take the slope of that line, and use that number to represent the sensitivity of the function at that point.

Let's finish with one more example. We'll continue working with the same function:

This point is called a local maximum, and we'll be able to calculate its coordinates later down the road.

If you've worked through the questions above, you now understand the concept of sensitivity. However, there are clear difficulties with the informal way we've been talking about sensitivity. For example, how would we actually calculate it?

Therefore, our next step is to define a derivative, which mathematically captures the concept of sensitivity and allows us to use it. As we take that next step, remember what a derivative really is: a quantity measuring the sensitivity of one quantity to a small change in another quantity.

The idea of "sensitivity" as intuition for the derivative is due to Deane Yang; see here.

(Video) Understanding the Definition of Derivative

by PatrickJMT

Here's a video by patrickJMT explaining derivatives and understanding their definition.

The definition of a derivative is: $$f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}$$

This probably looks like a jumbled mess of nonsense but we're going to explain everything shortly. First is the notation for a derivative: $f'(x)$ and said "f prime of x." The rest of the definition represents the slope of a tangent line at $x$.

Say we have a function $f(x)$, our tangent line at a point is the straight line that extends out from that point.

What we're interested in is finding some way to calculate the slope of this tangent line. We use a roundabout way involving limits to find this. Remember, to find the slope we need two points on the line. We only have one point, which is where the tangent and the function touch.

What we're going to do is find the slope of two points on the graph. So we added some addition points on our function in blue. We want to find the slope of them and our original point in red.

When you take two points on a graph and connect them, the resulting line is called a secant line.

We can find the slope of this secant line because we know two points. But who cares? We're interesting in the slope of our tangent line. The second point of the secant line is far away from our original point. What we're going to do is take a point closer to our original point.

We can find the slope of this closer secant line. But again we want the slope of the tangent line. Well let's make another secant line but with a point even closer.

The idea is as we get closer and closer to our original point, the secant line starts to resemble the tangent line. Since we can find the secant line's slope, we can use this to figure out the tangent line's slope.

So what we're doing is find the slopes of secant lines. Let's label them arbitrarily on our graph. The first secant line we made has a slope of $s_1$, the second $s_2$ and so on.

What we do is pick points closer and closer to $x$ and find the slopes of those secant lines. The key idea becomes: $$\begin{array}{ccc} \text{slope of}&&\text{limit of the}\\ \text{tangent}&=&\text{slopes of the}\\ \text{line}&&\text{secant lines} \end{array}$$

From here we just need to make this key idea mathematical. So let's just go back to when we had one secant line.

To find the slope of secant need to know the other point. Let's say that its some distance $h$ away from our original point.

So if we're originally sitting at the point $x$ and we go a distance of $h$ our new point will be $x+h$.

Now we can generically label our two points as $(x,f(x))$ and $(x+h,f(x+h))$.

So now the slope of the secant line becomes: $$\frac{f(x+h)-f(x)}{(x+h)-x}$$ We can simplify the denominator by adding the $x$ and $-x$ together: $$\frac{f(x+h)-f(x)}{h}$$

Now we want to define the slope of the tangent line by using slopes of secant lines. So we define the tangent slope as $f'(x)$. $$f'(x)=\ \ \ \ \ \ \frac{f(x+h)-f(x)}{h}$$

But we're not done yet. We wanted to take the limit of these secant slopes. We also wanted to take the points as they got closer and closer to our original $x$ value. As the points get closer the distance between them decreases. So our $h$ value, the distance between the points, gets smaller and smaller. So the limit is as $h$ approaches $0$, which means we take points until there's no distance between them. $$f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}$$ Look at that! We now have a formula that gives us the slope of a tangent line!