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I understand that a derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. This makes sense, since we would take the limit of the interval over which the derivative is computed as the interval tends to zero.

In other words, take the limit of the average rate of change as the interval over which the average is computed tends to zero.

But I read today that "differentiation is a method to find an exact value for the rate of change at any given value of $x$".

So is it a linear approximation or is it an exact value?

Thanks.

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  • $\begingroup$ The derivative is an object which is derived from the properties of a function. It may be interpreted as a linear approximation, but is an independent object in its own right. It makes no sense to talk about an "exact linear approximation" at a point where a function is not linear! $\endgroup$ – Joshua Pepper Mar 4 '14 at 20:22
  • $\begingroup$ To be more precise, the derivative itself is not a linear approximation, but a function assigning to each point of the domain the value (slope) of the best linear approximation of $f$ at that point. $\endgroup$ – Joshua Pepper Mar 4 '14 at 20:24
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    $\begingroup$ Exact value for the rate of change. The rate of change is approximated as the slope of a secant line. The limit of the slopes of the secant lines give the slope of the tangent line. (So, they're talking about something different than approximating a function using the tangent line.) $\endgroup$ – David Mitra Mar 4 '14 at 20:25
  • $\begingroup$ There are many "good" linear approximations. The best is uniquely determined, and its slope is given by the derivative. $\endgroup$ – Christian Blatter Mar 4 '14 at 20:28
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    $\begingroup$ Adding to the other comments, the concept is a lot easier to understand once you have knowledge of taylor series. The linear approximation is just the first two terms of the taylor series of a function. $\endgroup$ – user130512 Mar 4 '14 at 20:29
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I think this comes down mainly to a conceptual issue: Imagine that you have a way of approximating the behavior of something, call it a function. Now, imagine that your approximation gets more and more accurate the closer you move to this function. If you were infinitely close to the function, your approximation becomes infinitely more accurate (and thus ceases to be an approximation—it becomes exact).

We know the slope of a line between two points, $(x_1, y_1)$ and $(x_2, y_2)$, is $\displaystyle \frac{y_2 - y_2}{x_2 - x_1}$—which can also be looked at as the average rate-of-change of the function between those two points (or, the approximation of the function's rate of change between those two points). Now, the closer these two points are to one another, the more accurate your approximation will be.

Let's take two points on an arbitrary function. We'll call these points $(x, f(x))$ and $(x + \delta{x}, f(x + \delta{x}))$, where $\delta{x}$ is defined as being an infinitely small quantity. (That is, if the function is differentiable, these points are infinitely close to one another because they deviate by an infinitely small quantity).

The derivative is defined as the slope of the line "between" these two infinitely close points. That is... $$\frac{df}{dx} = \frac{f(x + \delta{x}) - f(x)}{x +\delta{x} - x} = \frac{f(x + \delta{x}) - f(x)}{\delta{x}}$$ ... or, the more usual... $$\frac{df}{dx} = \lim_{\Delta{x} \rightarrow 0}\frac{f(x + \Delta{x}) - f(x)}{\Delta{x}}$$ (That is, as $\Delta{x}$ moves infinitely close to zero, becoming infinitely small).

Since these points are infinitely close, your approximation of the rate-of-change of the function becomes infinitely more accurate—or, in other words, it becomes exact.

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    $\begingroup$ In most models of "infinitesimal", it doesn't "cease to be an approximation": your error is still a nonzero infinitesimal. $\endgroup$ – user14972 Mar 4 '14 at 21:17
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    $\begingroup$ What is $\delta x$ ? Is it a real number? Is not, what set is it an element of? $\endgroup$ – Thomas Mar 4 '14 at 23:02
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    $\begingroup$ @Hurkyl Which is why the usual NSA definition of the derivative is not an infinitesimal difference quotient but its standard part. $\endgroup$ – Michael Greinecker Mar 5 '14 at 7:01
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Answered here

Is line element mathematically rigorous?

and probably in many other threads. From the point of view of that answer, calculus is a collection of almost always inexact calculations that silently carry and propagate small approximation errors that are ignored, because those errors, even when compounded during the course of a computation, are provably negligible (that is, "infinitesimal", in one or more senses that can be made precise) compared to the finite quantities of interest that are the end result.

The statement

"differentiation is a method to find an exact value for the rate of change at any given value of x"

is misleading, in that differentiation not a way to "find an exact value" for a quantity that exists separately from differentiation; it is a definition of what was previously a vague pseudo-quantity (the "rate of change at the given value $x$"). If there were an independent definition of the instantaneous rate of change one could ask whether the derivative is equal to it exactly or not.

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  • $\begingroup$ ... but if we do this, it behooves us to develop a mental concept of "computations modulo provably negligible errors", and ultimately it is significant that we can do exact computations with this mental concept. $\endgroup$ – user14972 Mar 5 '14 at 0:02
  • $\begingroup$ That's what "provably negligible" means: that there exist a consistent formalism for actually neglecting those things (allowing to treat them as zero in computations, subject to some rules) rather than having to pass them along explicitly at every step. This does not impede having two different concepts of equality in play, only the finer of which is deemed to be truly "exact". $\endgroup$ – zyx Mar 5 '14 at 0:18
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One of the most basic methods of analysis is the ancient "method of exhaustion": in its most basic form, if $x$ is a number with the property that it's smaller than every positive number and bigger than every negative number, then $x$ must be zero.

Take a moment to digest exactly what this statement is saying, and to convince yourself of the obviousness of that fact.

This obviously generalizes to other numbers than zero. And it can be applied to more general ideas, especially when they can be quantified.

An example of a more general idea that can be quantified is that of a linear approximation to a function near $a$. You say you already know this so maybe you want to skip this section, but I want to explain it in terms of exhaustion.

We usually insist on it being exactly right at $a$, so it must be of the form:

$$ f(x) \approx f(a) + m (x-a) $$

For example,

$$ x^2 \approx 1 + m (x-1) $$

The value $m$ quantifies the approximation in some sense: we usually call it the "slope".

We note that $m=1$ is a better approximation than $m=0$, because:

  • For $m=0$, the error is $x^2 - 1 = (x-1)(x+1)$
  • For $m=1$, the error is $x^2 - (1 + (x-1)) = (x-1)x$

note that for every value "near" $1$; e.g. every $x \in (0, 2)$, the error for $m=0$ is worse than the error for $m=1$.

It turns out that every $m < 2$ is too small to be the slope of the best approximation, and every $m > 2$ is too big. Thus, the best approximation is at $m=2$: $m=2$ is exactly the right number to be the slope of the "best linear approximation".

And indeed, if we check the error, we get $x^2 - (1 + 2(x-1)) = (x-1)^2$, which is a smaller error for values of $x$ sufficiently close to $1$ than $m=0$ or $m=1$, or any other value of $m$.


Now, what might the phrase "rate of change of a function at a point" possibly mean? If we wanted the change between two different points, or the "average" rate of change over an interval, that's easy to understand.

But what does it mean to be the rate of change at a single point?

It is hard to give a "direct" definition that works in every case we might consider, but we can combine the method of exhaustion with our intuition of rates of change to give a definition of this phrase: any value that is less than the slope of the best linear approximation is clearly too small to deserve to be called the rate of change. Similarly, any value that is greater than the slope of the best linear approximation is clearly too big. Thus, we define the rate of change of a function to be the number in the middle: i.e. the slope of the best linear approximation, a.k.a. the derivative at that point.

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Another point of view on differential calculus is through its relation to integral calculus -- specifically, the fundamental theorem of calculus, which says

$$ f(b) - f(a) = \int_a^b f'(x) \, dx $$

Differential calculus can be seen as the business of attaching values to a point that quantifies behavior near that point. The simplest example of this is simply the value of a continuous function: if $x$ is near $10$, then $x^2$ is near $100$, for suitable meanings of the words "near" and "near". (the precise statement is the $\epsilon-\delta$ definition of limits)

To capture more information, we introduce the differential of a function. In this case, the differential of $x^2$ is given by $d(x^2) = 2x \, dx$. The value of this differential at $x = 10$ is $20 \, dx$.

For single-variable differentiable functions, $df(x)$ is always proportional to $dx$, which means we can define a function $f'(x)$ to be the constant of proportion: $d(f(x)) = f'(x) \, dx$. This allows us to talk about the derivative $f'(x)$ as an ordinary function, allowing us to defer trying to give a precise meaning to the notion of a differential.

The integral is the tool to extract "ordinary" information out of the "differential" information; the fundamental theorem of calculus above shows how the actual change in $f$ is recovered by the integral of $d(f(x))$, and therefore $d(f(x))$ is a sort of differential version of the notion of the "change of the value of $f(x)$".

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Differentiation yields the slope of a linear approximation of the function at a point... not the linear approximation itself.

The rate of change yielded by the derivative is exact, but if the function is not linear, the linear approximation obtained from this exact rate of change is only so accurate.

The concept becomes more clear when looking at multivariate differentiability, particularly when noting that differentiability is equivalent to stating the existence of an invertible linear map which essentially serves as a locally linear transformation of the coordinate space onto a neighborhood of the function evaluated at some point.

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