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I have seen a different definition of derivative, given by a Math SE user with very high reputation:

If $y=kx+f(a)-ka$ intersects $y=f(x)$ only once in the neighborhood of $(a,f(a))$, then $k=f’(a)$.

This makes me think about a different way to obtain derivative:

e.g. For $f(x)=x^2$, consider an arbitrary point $(a,a^2)$, then the system $$\begin{cases} y=kx+a^2-ka\\ y=x^2\\ \end{cases}$$ has exactly one solution.

Therefore, the quadratic equation $$x^2-kx-a^2+ka=0$$ has exactly one solution.

Hence, $$\Delta=k^2-4(-a^2+ka)=0\implies k=2a$$

As a result, by the above definition, $k=f’(a)=\frac{d}{da}a^2=2a$.

Of course, this method cannot always work, because it is difficult to take the condition ‘in a neighborhood of’ into account.

My question is,

For a function $f(x)$, what are the necessary/sufficient conditions for the above method to work?

Szeto
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  • The derivative is the slope of the tangent. – DynamoBlaze Jul 14 '18 at 07:04
  • @MalayTheDynamo That’s one of the definitions. The above definition is given by a 100k user on Math SE. But I cannot remember where I see it. – Szeto Jul 14 '18 at 07:07
  • Assume some small positive $\epsilon$ and write 'if it intersects only once between the interval $y=f(a)$ and $y=f(a+\epsilon)$' – DynamoBlaze Jul 14 '18 at 07:11
  • It's not a definition. It's a theorem. That being said, something's off about it. There are many lines which intersects $f$ only once in the neighborhood of $(a,f(a))$. How do you tell which one to use? (For instance, for $f(x)=x^2$ and $a=1$, the line $y=2x-1$ only intersects once in the neighborhood of $(1,1)$, but so does $y=5x-4$.) – Arthur Jul 14 '18 at 07:19
  • @Arthur hmmm... good point – Szeto Jul 14 '18 at 07:23
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    I think this idea is not only unhelpful; it is downright wrong. I doubt very much that anyone having some familiarity with curves, slopes, and derivatives would formulate such a thing. – Christian Blatter Jul 14 '18 at 07:29
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    The quotation must have been taken out of context. – md2perpe Jul 14 '18 at 08:46
  • Was it perhaps this post from 2015 by Omnomnomnom that you are trying to recall? – Brevan Ellefsen Jul 14 '18 at 08:58

1 Answers1

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The definition you mention is close to the ancient concept of a tangent line to a curve as given in the works of Euclid and extended by Apollonius and Archimedes. Their definition of a tangent line goes something like this:

Given a point on a curve, draw a straight line that passes through that point and has no other point in common with the curve. This tangent line touches the curve at one point while any other straight line passing through that point will intersect or cut the curve at one other point.

This definition was sufficient for the limited range of curves encountered by early mathematicians. For example it is sufficient for conic sections. It does not suffice for higher degree curves. For example, given a cubic polynomial, its intersection with a straight line is a cubic equation with three roots in general. If the line is a tangent line, then one of the roots is a multiple root and in the case of a double root, the third root gives another point of intersection. This case arises in doubling points on elliptic curves.

The usual $\,\epsilon-\delta\,$ definition of derivative can be interpreted geometrically in terms of a given point $\,G\,$ on a connected curve $\,C\,$ and the set $\,S\,$ of all lines passing through $\,G.\,$ Consider the function $\,F\,$ that takes any other point $\,P\,$ of $\,C\,$ and maps it to the secant line in $\,S\,$ uniquely determined by $\,G\,$ and $\,P.\,$ If the tangent line $\,T\,$ at $\,G\,$ exists, then it has the property that $\,F(P)\,$ is arbitrarily close to $\,T\,$ assuming that $\,P\,$ is sufficiently close to $\,G.\,$

Suppose that the set $\, S \,$ has the property that, if for any $\, \ell \in S \,$ the set of points $\, P \in C \,$ such that $\, \ell = F(P) \,$ is bounded away from $\, G. \,$ In other words, the line $\, \ell \,$ intersects the curve $\, C \,$ only once at $\, G \,$ and nowhere else in a neighborhood of $\, G \,$ determined by $\, \ell. \,$ Further suppose that this neighborhood is independent of $\, \ell. \,$ In other words, there is a disk $\, D \,$ centered at $\, G \,$ such that if $\, P \in D \setminus \{G\}, \,$ then $\, F(P) \,$ intersects curve $\, C \,$ at exactly two points $\, G \,$ and $\, P. \,$ In particular, this will always be true if $\, C \,$ is a strictly convex curve.

Define the set $\, E := (C \cap D) \setminus \{G\}. \,$ A natural question is does $\, F( E ) = S? \,$ If yes, then every line $\ \ell \in S \,$ is a secant line. This means that there is a minimum positive distance of $\, E \,$ from $\, G \,$ and thus $\, E \,$ is disconnected from $\, G \,$ which is contrary to supposition. Therefore, there is at least one $\, \ell \in S \,$ which only intersects $\, C \,$ at $\, G. \,$ If there is exactly one such line, then it is the tangent line $\, T \,$ at $\, G. \,$ But, for example, if the curve is $\, y = x^3 \,$ at $\, G \,$ the origin, then all lines with negative slope through $\, G \,$ intersect no other point of the curve yet the tangent line is $\, y = 0. \,$

Somos
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