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In introductory calculus, the derivative of a differentiable function $f$ at some point is often taught as being the slope of the tangent line to the graph of the function at that point.

My question is, isn't this circular reasoning? Tangents, after all, are defined with derivatives. To be more precise, a tangent line to a graph of a real function $f$ at some point $a$ in its domain where it is differentiable is defined by the equation

$y=f(a) + f'(a)(x-a)$

The slope of the tangent line is defined as the derivative. Hence the statement "the derivative of a function at some point is the slope of the tangent line to the graph of the function at that point" essentially reads "the derivative of a function at some point is its derivative at that point". The more inquisitive student not readily accepting the informal notion of a tangent line is thus left confused.

This leads to a more general point: why bother discussing "tangents" and "instantaneous rate of change" at all? Some instructors (like my calculus teacher) decide to throw in infinitesimals as well, to my bemusement(my calculus class is brilliant, but not quite at the stage where we can understand the theory behind hyperreal numbers). Why isn't the analytic definition sufficient? Namely, consider a function $f:I \rightarrow \mathbb{R}$ where $I \subseteq \mathbb{R}$ and $x_0 \in I$. If there exists a real number $L$ such that $\forall \epsilon>0$ $\exists \delta>0$ such that for all $x \in I$, $0<|x-x_0|<\delta \Longrightarrow |\frac {f(x_0) - f(x)}{x_0 - x} - L|<\epsilon$, we say $f$ is differentiable at $x_0$ and denote $L=f'(x_0)$, the derivative of $f$ at $x_0$.

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  • $\begingroup$ It isn't circular unless you leave the $f'(a)$ in $y = f(a) + f'(a)(x-a)$ undefined. It is entirely standard to defined $f'(a)$ as the limit of the difference quotient at $a$ (if it exists). Often, when teaching we have a geometrical, or physical interpretation, or motivation in mind, we try to bring this to the front, rather than the analysis, because many calculus students don't even see the definition of a limit. $\endgroup$ – James Feb 17 '16 at 18:05
  • $\begingroup$ You are confusing "mathematical" definitions with intuitive interpretations. The graphical representation of the above is a tangent. $\endgroup$ – Moti Feb 17 '16 at 18:18
  • $\begingroup$ math.stackexchange.com/questions/1053482/… $\endgroup$ – user301988 May 28 '17 at 23:20
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Would be a circular reasoning if the followed way were \begin{align} \text{Formal definition of tangent }&\longrightarrow\text{ Formal definition of derivative}\\ &\longrightarrow\text{ Formal definition of tangent} \end{align}

However, this is not the case.

In the sentence the derivative is the slope of the tangent, the word "is" should be understood as "is interpreted as". In the sentence the slope of the tangent is the derivative, the word "is" should be understood as "is defined as".

Formally, the slope of the tangent is $f'(a)$ (derivative at $a$). But only intuitively $f'(a)$ is the slope of the tangent; in other words, the derivative "can be thought" as being the slope of the tangent. Formally, $f'(a)$ is the limit $\lim_\limits{x\to a} \frac{f(a)-f(x)}{a-x}$. It turns out that the motivation for the formal definition of $f'(a)$ is the geometric notion of tangent. So, the followed way is

\begin{align} \text{Geometric notion of tangent }&\longrightarrow\text{ Formal definition of derivative}\\ &\longrightarrow\text{ Formal definition of tangent} \end{align}

We are not working with multiple definitions. We are working with multiple ways of thinking. From this point of view, the reasoning is not circular.

The derivative can be thought of as:

  • Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.

  • Symbolic: the derivative of $x^n$ is $nx^{n-1}$, the derivative of $\sin(x)$ is $\cos(x)$ the derivative of $f\circ g$ is $f'\circ g*g'$, etc.

  • Logical: $f'(x)=d$ if and only if for every $\epsilon$ there is a $δ$ such that when $0<|\Delta x|< \delta$, $$\left|\frac{f(x+\Delta x)-f(x)}{\Delta x}-d\right|<\delta.$$

  • Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.

  • Rate: the instantaneous speed of $f(t)$, when $t$ is time.

  • Approximation: The derivative of a function is the best linear approximation to the function near a point.

  • Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power.

This is a list of different ways of thinking about or conceiving of the derivative, rather than a list of different logical definitions. (W. P. Thurston, On proof and progress in mathematics)

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  • $\begingroup$ Thanks, this is a helpful response! It's just, the thing is, some instructors at the introductory level do not explicitly highlight the distinction between geometric/intuitive ideas of what $f'(a)$ means and the formal definition. For example: consider the Wikipedia page on the derivative. It states in the second paragraph that the "derivative of a function of a single variable at a chosen input value...is the slope of the tangent line to the graph of the function at that point". Wouldn't it be more precise to replace the is by "can be conceived of as"? $\endgroup$ – MathematicsStudent1122 Feb 18 '16 at 16:08
  • $\begingroup$ @MathematicsStudent1122 I agree with you. For a clear understanding, the distinction between the formal discussion and the informal discussion should be considered. Once we have done it, we can abuse the language to make the communication easier. However, If no clarification is done, doubts emerges. $\endgroup$ – Pedro Feb 18 '16 at 16:19

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