# Why is the derivative important? [duplicate]

Derivatives, both ordinary and partial, appear often in my mathematics courses. However, my teachers have never really given a good example of why the derivative is useful. My questions:

1. Other than the usual instantaneous rate of change, what are some common uses of the derivative?

2. What does the partial derivative tell us? And what does the total derivative tell us?

I find that often times, the derivative is simply explained as "the instantaneous rate of change".

I am thinking about switching my major, because the applications of math at such an elementary level seem trivial when professors just push symbols and don't have any real world motivation included in their lectures.

P.S. This question is not a duplicate of Why do we differentiate? I do not want to know why we differentiate. I want to know why it is important past our undergraduate learning. What are the applications beyond Calculus 3? Beyond academia, what makes the derivative important in complex situations?

• @MichaelByrne Because it is a tool that solves important problems. I don’t love math because I love derivatives—I love it because it gives me tools (like derivatives) to solve important problems. For example, I work with people who use optimization models to improve decision making in healthcare. We use derivatives, along with hundred of other tools, every day. For me the spark didn’t really come taking classes (I agree lots of “symbol pushing” in undergrad). It was when I found an important problem that seemed to have nothing to do with math and I realized I needed a derivative. Jun 12, 2018 at 23:58
• Most of our modern physic is based on differential equations. Derivative is also a very usefull tool to find minimum/maximum of functions. And this has huge applications in the real world Jun 13, 2018 at 0:04
• If you are so concerned with real world applications then switching majors is probably a good idea. Jun 13, 2018 at 5:29
• acceleration is the derivative of speed. In the last centuries, we have worked a lot on it : from cars to rockets. Jun 13, 2018 at 6:36
• Hmm if there is any part of mathematics that has practical applications, I would say it is derivatives. What world are your professors living in that they cannot give you any useful example? Jun 13, 2018 at 7:53

The derivative has many important applications both from elementary calculus, to multivariate calculus, and far beyond.

The derivative does explain the instantaneous rate of change, but further derivatives can tell the acceleration amongst other things.

With optimization, the derivative can tell us where the best place to sit in a room is, if the room is filling up with smoke, and at what time it is the best to sit there. The derivative can help with many optimization problems.

The partial derivative tells us the direction of variables at a given time and the total derivative tells us where the slope increases the most and where. This is one way we can optimize in $\mathbb{R}^3$. The derivative can be applied to water flow and generally tells us much about how things change with respect to another variable.

The derivative further can help in industry with economics, healthcare, engineering (especially), and many other things. Business has many applications as well. Your professor might not have time to delve into these applications as much as you would like because it is a calculus class, not an "application of the derivative" class. Although, he should definitely discuss these issues at some point. I have had some professors in my time who glossed over such subjects, but in multivariable calculus, they go way more in depth with them. I don't suggest switching your major without speaking directly with your professor about your difficulties.

If you have further questions, I encourage you to ask your professor in office hours the same exact question and voice your concerns there. A good professor will encourage and motivate your learning outside of the classroom if you show initiative and ask.

• I appreciate your insight. Can you perhaps tell me more about how your professor introduced the derivative?
– user568999
Jun 13, 2018 at 0:24
• Sure thing. My professor introduced the derivative by explaining how in algebra, we cannot attain the instantaneous slope. We can get very close, but it would still be a tangent between two points. With calculus, we can get infinitely close, so much so, that they are one and the same, essentially. We then covered applications and optimization. (i.e. how to build a barn with limited constraints). Most professors go about it this way, or at least, most to my knowledge, including mine, did. As others have stated, the applications to physics and engineering are enormous. Jun 13, 2018 at 0:27
• It might be worth pointing out that derivatives are essential to ODE's and PDE's and most/all the numerical methods used to model pretty much any non-trivial, physical system.
– DRF
Jun 13, 2018 at 11:00
• Most real-world places where you ask "If I change this, how does that change?" of some numerical quantity are best considered using derivatives. That's a lot of places. Also, lots of physical quantities (momentum, velocity, power, torque, ...) are the derivatives of (combinations of) others. That's the two main reasons why they're so ubiquitous it seems to me.
– user569550
Jun 13, 2018 at 11:29

I'm going to take a slightly different tack than most of the other answers here and point out that "important" (the word used in the title of the question) and "useful" (the word used in the body) are not exactly synonyms. Something can be important in different ways:

• It may be important because it is useful for solving real-word, practical problems
• It may be important because it is useful for solving theoretical, non-applied problems
• It may be important for historical reasons
• It may be important because it is surprising or counterintuitive
• It may be important because it illuminates a mystery

A lot of these might be summarized by saying that something is "interesting". I think of "interesting" and "useful" as orthogonal axes of value, in the sense that they are two completely independent ways of saying why something is worth knowing.

Most mathematicians, I would venture to say, are motivated by things other than "practical applications". (Some even actively scorn applications, although I do not go that far.) According to tradition, when Euclid was asked "Why is this useful?", he replied sarcastically

"Give him threepence [lit: a three-obol piece], since he must make gain out of what he learns."

The derivative -- and calculus in general -- is important and interesting in many of the above senses, quite apart from the practical applications (which, it must be said, are extremely abundant). From the days of the ancient Greeks to the time of Newton and Leibniz, philosophers struggled to understand the nature of motion itself, which many of them regarded as fundamentally paradoxical: if, in any given moment, zero time elapses -- and therefore in any given moment an object's position does not change -- how is motion possible? More generally, how do we get the experience of smooth, continuous change from a sequence of infinitely many distinct points in time? (The geometric version of this is: if a point has zero size, and a line is just a set of points, how do lines have size?)

People have thought of these questions as interesting for centuries because they are head-scratchers. They lead one to ponder the infinitely many, the infinitely small, and the ways in which infinities can balance each other out to produce finite quantities. That stuff is cool, and Calculus provides a set of techniques for figuring it out, and a language for talking about it coherently.

Once you formalize it, derivatives also allow you to discover things that are genuinely surprising, for example that it's possible to have a curve that is continuous everywhere but not differentiable anywhere. (That statement doesn't even make sense without derivatives, but if you understand derivatives you can begin to appreciate how absolutely baffling and nonintuitive such a thing must be.) The capacity to be surprised and the ability to contemplate the infinite are part of our sense of wonder, and essential components of being human. I would say that's pretty important, whether or not it's "useful".

• A quote from Hardy: "I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." (He was quite wrong, unfortunately for him.) And an anecdote: Titchmarsh writes in his obituary "I worked on the theory of Fourier integrals under his guidance for a good many years before I discovered for myself that this theory has applications in applied mathematics, if the solution of certain differential equations can be called 'applied'. I never heard him refer to these applications." Jun 13, 2018 at 21:53
• And while most mathematicians wouldn't go to that extreme, the attitude certainly survives. Jun 13, 2018 at 21:53

The derivative locally measures how much a function stretches its domain at a point. If it is negative, there is both stretching and reversal of direction.

• Derivatives....stretch?
– user568999
Jun 12, 2018 at 23:56
• Yes. Look at a short sequence of points near $x = 3$. Now look at their squares. They will be roughly 6 times as far apart. The derivative in many dimension does basically the same thing as it is a linear transformation. That a derivative is a numerical function in one dimension is an accident. Jun 13, 2018 at 0:00
• @MichaelByrne Let 3Blue1Brown explain it to you.
– SQB
Jun 13, 2018 at 6:12
• I really came to understand this when I first read Michael Spivak's Calculus on Manifolds. Jun 13, 2018 at 11:57

Have you ever got a ticket for speeding? If yes, your derivative was higher than legally allowed.

Derivative is used in finding rate of change, slope of tangent, marginal profit, marginal cost, marginal revenue, linear approximations, infinite series representation of functions, optimization problems, and many more applications.

Partial derivatives, directional derivatives, total derivatives are concepts of multivariable calculus and are used in optimization and linearization of multivariable functions.

• You know what I meant. ( speed = derivative ) Jun 13, 2018 at 18:40
• best answer, since as down to earth as possible (+1) Jun 13, 2018 at 18:52

I expect lots of excellent answers to this question will come up. I'll chime in with just one interesting aspect of the derivative, that became evident to me relatively late in the study of calculus.

When the derivative is $0$, you are generally at some sort of extremal point. Maximum, minimum, optimum. You are at the top of the hill, you have the optimal portfolio, your bridge is standing still and your chi is in balance. It's often the stopping point of your search, either because you've found what you wanted, or because you have no (obvious) direction to go.

When the derivative is not $0$, you have "room", and stuff behaves "normally". If you are looking for something, you have a clear direction to take. Crucially, if your derivative is not $0$, your function is locally invertible, and all sort of things become possible -- lots of theorems have "the derivative (or its multidimensional equivalent, the jacobian's determinant) is not $0$ at your point of interest" in their hypotheses exactly for this reason.

Problems tend to crop up when the derivative (or its multidimensional equivalent, the jacobian's determinant) is somewhere in between: very small, but not quite $0$. These are situations many applied mathematicians loathe, because you are not at the end of your journey, but you have very little room or clarity of direction left. Imperceptible perturbations can create lots of problems. You'll hear often the term "ill-conditioned" in these cases, and one recurrent problem is to decide when your derivative is close enough to $0$ that it can be treated as $0$.

While all the other answers highlight useful aspects of the derivative, I feel that most of them are a bit vague and don't mentioned which problems exactly, concerning mathematics, not real-world applications the concept of derivative allows you to solve.

Now, in the context of a single function, the concept of derivative allows you to
1) find minima or maxima of differentiable functions
2) approximate differentiable functions by simpler functions (e.g., by Taylor expansions)

There are a number of real-world application were 1) and 2) are essential - but there are also applications inside of mathematics, but fewer.

But I think the true power of the derivative arises, when you look at the possibilities derivative gives you, when you consider a whole of functions.
Namely in this case you can construct important models, by formulating equations that contain the derivative. Consider $$\frac{\partial u}{\partial t}= \frac{\partial^2 u}{\partial x^2}.$$ Here we are consider a set of all sufficiently differentiable functions $u$ of the form $u:[0,\infty)\times\mathbb{R}\rightarrow\mathbb{R}$.

Why is this equation useful? This is the one-dimensional heat equation (the wikipedia link is a bit more complicated). By appealing to arguments from physics about rate of change (hence the derivative) of heat flow, this equation is obtained! If we didn't knew about the concept of partial derivative $\partial$, we would not even be able to derive the equation. (But thinking about of rates of change long enough will make you discover the concept of derivative - this is actually how Newton discovered it back in 'ol 17th century.)

This equations models how heat flow through a one-dimensional object (e.g. a rod) with time. (Usually things are more complicated, because one also specifies additional constraints in a second equation, such as what temperature we should start with; which is called "initial condition" but it is a technical condition which we shall skip here). If we are able to solve this equation, than we can model heat flow. Isn't that neat?!

Now, this wasn't just some isolated equation were the derivative turned out to be necessary to even formulate the problem; the paradigm "use arguments from physics which involve rates of change of some quantity -> derive an equation about those quantities which will involve derivative -> solve it to understand how those quantities change" is ubiquitous in engineering and physics.

For the second question, let think geometrically, in 2 dimensions. The partial derivative at a point $x_0$ means reducing your problem to a derivative in one dimension. The graph then looks like a surface. If you take a slice through that surface on a line parallel to the $x$ or $y$ axes that goes through that point $x_0$, you obtain a function in 1 dimension. The derivative of this one-dimensional function at $x_0$ is called "partial derivative".

In contrast, the total derivative at $x_0$ is the proper generalization of the one-dimensional derivative, not just a reduction like the partial derivative. Geometrically, the one-dimensional derivative is the slope of the tangent at $x_0$. The total derivative is then a collection of slopes (which you collected in something you call gradient, or more general, Jacobi matrix) of a tangent plane.

All of these geometric picture can be algebraically generalized from 2 dimensions to $n$. There you don't have any geometric intuition, but you don't need to, since things work analogous to two dimensions.

A concrete though frivolous example:

A hobby of mine is designing abstract sculptures for 3d printing. Each is generated by a program that I write for that purpose. A tangible object can't be an abstract surface of zero thickness, so the program needs to know the orientation of the surface, i.e., what direction is away from the surface (because it wouldn't help to offset within the surface); this I get from the partial derivatives of the coordinates with respect to the parametric variables. The second derivative tells me how strongly the surface is curved, and thus how closely the sample points need to be spaced to keep the result accurate to within the resolution of the process.

Suppose we have some unknown function $f(x)$ and we're trying to find a polynomial that approximates it, of the form $g(x) = \lambda_3 x^3 + \lambda_2 x^2 + \lambda_1 x + \lambda_0$. In that case, we want to find the set of $\lambda$s that minimizes the difference between $f(x)$ and our polynomial. So we define an error function, $E(\lambda) = \sum_x (f(x) - g(x))^2$. It just sums up the error over all values of x (or a random sample drawn from the set of the possible values, in practice). We square the difference so the negative and positive errors don't cancel out; it's basically just a way of taking the absolute value while keeping the function differentiable.
Then the gradient would be $\Delta_E = [ \frac{\partial E}{\partial \lambda_3}, \frac{\partial E}{\partial \lambda_2}, \frac{\partial E}{\partial \lambda_1}, \frac{\partial E}{\partial \lambda_0}]$
We then start with random values for each $\lambda_i$. (Or equivalently, we start at a random point $\lambda$ in a 4-dimensional space.) We can approximate the gradient at that point by sampling the error at nearby points. (Note that we aren't actually taking the derivative analytically; we're measuring the slope.) Since the gradient gives us the rate of change of the error function, we can use it to find the steepest slope at our current position. We move our $\lambda$ vector a small distance in that direction, then repeat the process. Eventually, we end up in a spot where the derivative is approximately zero, called a local minimum. (If any of that's unclear, googling "gradient descent visualization" should turn up some useful videos of the process).