What does the derivative really mean?

I was introduced to calculus a few weeks ago, and while I can "solve" problems consisting of derivatives and integrals, I still do not truly understand what the derivative means. Here are some of the arguments/explanations I have heard :

1. Derivative is the instantaneous rate of change.

However, this to me never makes sense, for a change to occur there needs to be an interval, but if there is an interval, then it is not instantaneous?

2. Derivative is the slope of the tangent line.

This is simple and easy to understand, but it I do not understand how the "slope" of the tangent line tells us how "fast" the function is changing at a point.

3. Derivative is the sensitivity of the function at the point.

This to me, is the most appealing definition, at a point, the derivative measures how much my function will change around that point if i make a tiny change in my input variable. However, this still causes a bit of confusion to me. How does this "sensitivity" intuition lead to the limit definition of the derivative?

I am sorry, if I made any conceptual errors in my interpretations of above definitions. It would be a great help if someone could help me understand derivatives better.

• Ultimately what the derivative actually means is its precise limit definition, and nothing else. Any other explanations you see are just imprecise intuitive approximations to that meaning. Jan 23 at 6:33
• You can try to watch this video from YouTube as it discusses your point where the derivative is an instantaneous rate of change. Jan 23 at 6:35
• You may enjoy reading Feynman's chapter on motion, at least the first three sections. It is packed with more insight than the entertaining style might suggest.
– dxiv
Jan 23 at 6:38
• Suppose you’re driving and you look at the speedometer at a particular moment in time and it says 10 meters per second. In that instant, your velocity is 10 meters per second. Instantaneous velocity is defined as the limit of the average velocity as the duration of the time interval over which we are computing the average velocity approaches $0$. Jan 23 at 6:39
• In my opinion the best intuitive description of the derivative is as the best linear approximation to the function at the given point. There's an interesting bit of "mathematical hermeneutics" here, of course: somehow what's going on is that we start with multiple very simple but somehow flawed intuitions, and then as we develop a precise definition we also develop a more accurate - if more involved - intuition. Jan 23 at 6:41

3 Answers

Since you like "sensitivity" the best, one way to think of the derivative is "how much the function stretches or shrinks a small interval." Suppose the derivative of $$f(x)$$ at $$x=x_0$$ is $$1/3$$. If you take a small interval $$(x_0-\delta,x_0+\delta)$$ which has width $$2\delta$$ and put all those values into the function, then their image (on the $$y$$-axis) is another small interval. The width of that interval will be about $$(1/3)2\delta.$$ The smaller the interval on the $$x$$-axis, the closer the width of the interval on the $$x$$-axis will be to exactly $$1/3.$$ The derivative tells you that little neighborhoods of $$x_0$$ get shrunk by a factor of $$1/3$$ when you push them through $$f$$.

• Thank you a lot, it cleared a lot of things. Jan 24 at 12:58

This is an excellent question - particularly from someone only a few weeks into calculus.

Differentiation is an attempt to understand things that change a variable rate. It starts with the simple idea illustrated by the formula $$\text{speed} = \frac{\text{distance}}{\text{time}}.$$ The graph of the function that tells you how the distance you cover when traveling at a constant speed depends on the time you've traveled is a straight line whose slope is your speed.

When you are not moving at a constant speed then the problem is harder. A falling stone falls faster and faster. The graph of the distance versus time function is a curve that gets steeper and steeper as time goes on. At any point the slope of the tangent tells you how fast the stone would be falling if gravity suddenly stopped working. Acceleration would cease but the stone would not sop in midair. The distance function would follow the tangent line from that point on.

That begs a question: does the falling stone even have a speed at any particular instant? That's Your question (1) and it puzzled the Greeks long before calculus. It's one of Zeno's paradoxes.

The practical answer is that Newton had to be able to reason correctly about instantaneous velocity in order to invent his physics, so he thought of it as the average velocity over an infinitesimally small time interval. He never clarified what "infinitesimal" meant, and his contemporaries criticized him for that. Today we avoid infinitesimals by defining the instantaneous velocity to be what the average velocity is close to when you calculate the average over small but real time intervals. That's the "limit of the difference quotient" in your textbook. (Of course that just pushes the philosophical question off to understand what "close to" and "limit" mean. Some beginning calculus courses address that, some postpone it to more advanced courses.

At a more philosophical level, calculus is "really" several things. Imagine the answers to "what are numbers and arithmetic, really?" Useful tools for commerce. Rules for solving problems. A formal system with axioms and theorems. For some people, beautiful patterns.

You can give the same kinds of answers for calculus. It was invented for physics. There are rules for getting the answers. It has definitions and theorems. It's beautiful.

• It indeed is beautiful. Thank you for helping me with my doubt. Jan 24 at 12:59

Here's a helpful sequence in which to define things (once you already know what limits are, as the second bullet point below needs them):

• For $$h\ne0$$, a secant from $$x=a$$ to $$x=a+h$$ on $$y=f(x)$$ is the straight line joining $$(a,\,f(a))$$ to $$(a+h,\,f(a+h))$$. Its gradient is uncontroversial.
• The $$h\to0$$ limit of the secant's gradient is the derivative at $$a$$, denoted $$f^\prime(a)$$, or $$\left.\frac{\mathrm dy}{\mathrm dx}\right|_{x=a}$$. We can do this once for each choice of $$a$$, thus obtaining a function that's the derivative of the original one. To take a no doubt familiar example,$$y(x):=x^2\implies\underbrace{\left.\frac{\mathrm dy}{\mathrm dx}\right|_{x=a}=\lim_{h\to0}\frac{2ah+h^2}{h}=2a}_{\forall a}\implies\frac{\mathrm dy}{\mathrm dx}=2x.$$
• The tangent at $$x=a$$ is the line through $$(a,\,f(a))$$ whose gradient is the derivative at $$a$$.

Now let's go through your numbered options:

1. In the early history of calculus, your first objection occurred in a criticism of differentiation that used the phrase "ghosts of departed quantities". In terms of the above bullet points, the answer is that the derivative is a limit of secant derivatives, so we define the tangent in terms of that, rather than doing the tangent first.
2. This fact is a corollary of our tangent-last approach, not how we define the derivative in the first place.
3. How would you quantify how quickly $$f$$ changes at $$a$$? If $$f$$ varies from $$x=a$$ to $$x=a+h$$, the secant gradient will encounter the objection, "but it doesn't undergo a linear change between these values of $$x$$; the function is not its own secant". But if the $$h\to0$$ limit exists, that limit is arbitrarily well-approximated by secant gradients when $$h$$ is small enough. Were it not so, speedometers wouldn't work.
• Thank you, it was very helpful! Jan 24 at 12:56