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:

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

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?

SincerelyPrime 06/13/2018.

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.

mweiss 06/13/2018.

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".

ncmathsadist 06/12/2018.

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.

Mohammad Riazi-Kermani 06/12/2018.

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.

Anonymous 06/13/2018.

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 $0$.

user10324 06/13/2018.

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.

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