@Pat: Thanks, glad you liked it! Oh man, how I wish I could go back in time and give myself some tutorials :).
@Zaine: Thanks, I really appreciate it!
Wow, now that’s power. Takes a brilliant mind to break complexity down, making this one of the best sites online!
Thank you Tanksley, for your explanation. But I have to admit, there is a lot in the above explanation that I am not familiar with( like the first order semiderivative ) , so I’ll have to go through it step by step. I hope you’ll stay on the site to clarify my doubts!
In the mean time, can we discuss your earlier comment “The derivative is sufficiently understood as the slope of the line tangent to a curve at a point.” ?
Let’s say we want draw a tangent to a curve. This raises the question what is a tangent.
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Let’s say the tangent a point is a line that best approximates the curve at the point. This raises the question what is meant by best approximation ?
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A simplified answer to this question would be that it should have the same value at the point as the curve.
So if your function is y = x^2. The then at x = 2, y = 4. But you can draw any number of lines through the point (2,4). So how do you go from there?
For a circle or a conic you can draw the a line from the centre or the focii and then define the tangent as the line that is perpendicular to this line. But how would you draw a tangent to an arbitrary curve (one that has no centre or focii)?
I would also like to know if my statements 1 and 2 are correct, or do they need some mathematical refinement.
I just wanted to let you know that I really appreciate the effort you put into this. I only discovered this website a few days ago, and I’ve been having a blast reading all those intuitive approaches!!
You should consider writing an elementary and highschool book of mathematics, as well as teaching on khansacademy 
Please keep this flowing 
and if there’s any way we, the audience, can support you, please do mention how!
(Note: I hope the LaTeX below works. I wish there were a preview mode…)
I claimed that using limits and infinitesimals to define the derivative led to restricting ourselves to functions, while using the geometric definition of the derivative allowed arbitrary curves rather than only functions. (There are other advantages; for example, using the geometric definition allows you to reason about derivatives of curves over arbitrary fields rather than only the continuum.)
Recall that the geometric definition of the derivative is the slope of the line tangent to the curve at any point on the curve. First let me distinguish a “function” from a “curve”. Every function is a curve, but a function has at most one value per input, while a curve can have any number of values. We can consider the subset of general curves called the “algebraic curves”, consisting of the Cartesian graphs of the polynomials of the appropriate number of variables for the dimension we’re examining; analytic curves are also amenable to this analysis, or curves on other coordinate systems.
And a simple example of that is the classical unit circle. In order to find the derivative of the unit circle using limits, one has to split the circle into upper and lower halves. If one uses the geometric definition, however, there is only one curve, and computing a formula for its tangent line is simple algebra. The result is a formula for the tangent line to the circle at every point on the plane (sometimes called the “first order semiderivative”), and it’s easy to see how to extract the slope of that line.
The algebra one performs in order to extract this is to evaluate the curve at $$(x+r,y+s)$$, where r and s are variables representing arbitrary numbers, then express the result in terms of powers of x and y, and finally evaluate that at $$(x-r,y-s)$$, thereby giving a net effect of adding and subtracting zero and rewriting the expression in terms of powers of $$x-r$$ and $$y-s$$. (This action substitutes for adding and subtracting an infinitesimal, but we need no assumption that infinitesimals exist.) If the original curve was algebraic it will also be analytic, and so the rewritten result will be a Taylor expansion.
Now, to find the slope of the tangent line, one needs only to see that the equation of the tangent line is the equation setting all the zeroth and first order terms in the Taylor expansion to zero (and discarding all the higher order terms); and the equation of the slope of that line is simply the coefficient of $$x$$ divided by the coefficient of $$y$$.
So, let’s compute the first order semiderivative of the unit circle.
The curve is $$x^2+y^2-1$$. Evaluating at $$(x+r,y+s)$$, we get the translated curve $$(x+r)^2+(y+s)^2-1$$, which expands to $$x^2+y^2+2rx+2sy+(s^2+r^2-1)$$. The Taylor expansion is therefore $$(x-r)^2+(y-s)^2+2r(x-r)+2s(y-s)+(s^2+r^2-1)$$.
To find the equation of the tangent line (the first-order semiderivative with respect to x and y), we set the zeroth and first order terms of the Taylor expansion equal to zero: $$2r(x-r)+2s(y-s)+(s^2+r^2-1)=0$$. Putting this in the standard y=mx+b line, we get $$y=(-r/s)x+(s^2+r^2/2)/s$$ as the equation of the line tangent to the unit circle at (r,s). Therefore, the derivative of the unit circle curve at the point $$(r,s)$$ on the circle is $$(-r/s)$$ for all points where $$s \neq 0$$.
This follows directly for all algebraic curves, and can be confirmed for all analytic curves. For non-analytic curves, it can be shown that we can approximate the derivative as closely as desired.
-Wm
Nikhil said: “But I will first wait for your comment on the application of the derivative to arbitrary curves and how the limit restricts this applicability.”
Thank you for reminding me that I said that – I forgot to explain that part.
I just explained how to apply the geometric definition of the derivative to arbitrary algebraic curves. More complex curves are also available, and there are proofs that the geometric definition yields both exact solutions and a simple method for deriving approximations.
I also explained one obvious way in which the geometric definition is superior, in that it allows derivatives of curves that aren’t simple functions. But I didn’t explain in what aspects the infinitesimal definition of the derivative is inadequate. Notice that I’m not trying to say that it’s bad or wrong, or that it’s ALWAYS inadequate; rather, I’m pointing out some specific problems that hinder certain uses. Also notice that I’m not complaining about limits; I’m talking specifically about the use of infinitesimals in the definition of the derivative. Limits may still be useful (for example, I mentioned piecewise smooth functions, whose derivatives require limits).
The most interesting problem is that infinitesimals require the use of the continuum, and not all numbers are embedded in a continuum. The rationals are very useful for most purposes; and floating point computation is a use of a special type of rational number. There are other infinite fields as well, and obviously the finite fields cannot be approached with limits at all (but are quite easily approached with geometry). And yes, the definition of “algebraic curve” applies over any field, finite or infinite, so this method will find its derivative. Complex numbers are reachable as well – in fact, you can probably see that the equation I derived for the tangent line has values over the entire plane, not just on the unit circle, and in fact those values are geometrically meaningful.
There are more interesting results as well. The tangent line is interesting and useful, but there are also tangent conics, cubics, and so on.
-Wm
Sorry about the LaTex. Ugly.
Khalid,
This is great! Derivatives were always out of focus to me but this is helping clear things up.
Sebastian
“You said that I mentioned infinity and the continuum. I didn’t mention either; the only place I can find those concepts is in the original post.”
Yes I was referring to the original post. I just stumbled upon this article while doing a google search. and assumed that you were its author. Now I have explored the site, and discovered it was Kalid.
I was thinking about your comment - “The derivative is sufficiently understood as the slope of the line tangent to a curve at a point.”. I have some doubts regarding this definition. But I will first wait for your comment on the application of the derivative to arbitrary curves and how the limit restricts this applicability.
Nikhail, you said “To properly understand a derivative you would need the concept of a limit.” That’s the sentence I was seeking to correct. Your last paragraph claims that you don’t need limits but then implies that you need infinitesimals, and this is also something I disputed – but assuming your post is not self-contradictory, your claim would imply that you need infinitesimals in order to understand derivatives improperly, and if you add limits you can understand them properly, and there’s no other way to even begin to understand derivatives.
I contradicted this claim by saying that there is another way of understanding the derivative: the geometric definition. It requires no limits, no infinitesimals, no continuum. It works not only on smooth functions, but also on arbitrary smooth curves. (I’ll explain in my next comment.)
You said that I mentioned infinity and the continuum. I didn’t mention either; the only place I can find those concepts is in the original post. I would also disagree entirely with the original post’s take on them; for example, there is not only one infinitesimal, rather, there are an unknown number of them, so you cannot iterate through the continuum by adding just any infinitesimal to a number (if you do this, you’ll miss points on the continuum).
On the other hand, I do agree that one does not need epsilon-delta to introduce limits; one can introduce limits for other purposes. Or one can introduce limits for their own sake. But this has nothing to do with the topic of understanding the derivative.
-Wm
@wm: Thanks for the pointer! I’ll check it out.
Hi Khalid,
Great article. I have always been fascinated by calculus and always wanted to decipher the true meaning of derivative. Your article gives me a great insight. However I would beg you to clarify the following confusion that has arisen.
We all know that derivative of Y = X^2 is 2x. when you calculate values of y for x=2 and 3, you get y = 4 and 9 respectively. The change in y here is 9-4 = 5. However if I substitute x= 2 in the derivative function dy/dx it gives me 2x = 4. you showed us why this difference exists. It is because of the dx factor (Shoddy instrument). But the reality is that y changed by 5 units when x changed from 2 to 3. Are you saying that dy/dx or derivative is not here to calculate rate of change for such large changes and if you use it for large changes results are inaccurate. Does that mean that dy/dx can only be used to calculate very small changes.
Earlier I thought if you want to find how a function f(x) is changing w.r.t x between 2 values without substituting the values, just calculate the derivative and substitute x but it seems I was wrong?
Also I didn’t understand when you say
The derivative is 44" means "At our current location, our rate of change is 44."
Change is a relative term. How can there be a change at a current location. It has always got to be between two locations.
@Still learning: Thanks – really appreciate the encouragement!
Thanks Robin, I really appreciate the note. I love it when people in other fields are able to take away some insights. I definitely think math pedagogy needs to change, to use other techniques, but really, to just ask “Are we actually learning here?”. I feel there’s a giant emperor’s clothes problem where nobody wants to admit “Hey, this concept we’re supposed to be teaching… it’s not clicking at an intuitive level, and it should.” 3d visualizations and other tools can help get ideas to really sink in. Appreciate the note!
Tanksley , you say that the concept of a limit restricts the definition of a derivative to functions and makes it inapplicable to arbitrary curves. I did not follow this. Could you elaborate ?
" Nikhil, you do not need limits or infinitesimals to properly understand the derivative. "
I never said that you need limits to understand the derivative. Read the last paragraph of my comment.
In your explanation, you talk about infinity and the continuum. What I was saying is - that the conceptual leap from there to that of a limit is very small. So there is no need to avoid the concept of a limit.
One doesn’t need the epsilon - delta definition to introduce the concept of a limit.
This is a fair explanation of the theory behind derivatives; but I like how Wilberger explains and motivates tangent curves (which are directly and simply related to derivatives). Not only does he NOT use the idea of “dx” (which doesn’t actually exist in any system of numbers beyond the integers, since there is no unique number that is closest to zero), but he winds up defining the theory so that it works on arbitrary algebraic curves (not only functions).
Check it out – look at his (njwilberger’s) Math Foundations series on YouTube. Most people reading here will be able to skip to something like the episode on doing calculus on the unit circle, but don’t expect to understand EVERYTHING if you do that. The interesting thing is that he defines this without using limits at all; the essential point is that he uses “the nth degree polynomial that best approximates the surface at that point” (of course, this is the Taylor expansion at that point).
-Wm
@Bassman, Ogbuka: I’ll take those as suggestions for future topics, thanks.
@Asmaul: Glad it was helpeful!
@John: Thanks for the note, really appreciate it! I hear you, so many math explanations just focus on the grammar, like the lifeless language classes that nobody ever seems to learn from (contrasted with learning a language by actually being immersed in it and speaking it, vs. trying to crunch through the rules like a computer).
I’m going to update the article right now with the new integral/anti-derivative analogy. Thanks again for posting!
Really great stuff. Mathematics is the foundation of all science and science is the compass to help us navigate the universe. Keep up the good work. Very much appreciated.
Nikhil, you do not need limits or infinitesimals to properly understand the derivative. The derivative is sufficiently understood as the slope of the line tangent to a curve at a point. This geometric understanding does not invoke limits or infinitesimals. You can add in limits to this definition to handle piecewise continuous curves, but as-is this definition can handle arbitrary curves, rather than being limited to functions.
If one is learning general calculus then infinitesimals are essential; but if one is learning the derivative they are not, and therefore no limits are needed. Iverson actually wrote a Calculus text without using limits, and he only used infinitesimals informally. It’s available online at http://www.jsoftware.com/jwiki/Books. Aside from that oddity, the text is notable for its computational focus and for its treatment of some advanced theoretical topics such as fractional integrals (Wikipedia calls this the “differintegral”).
-Wm