Ben Fowers Semester Project

The History of Derivatives

Introduction:

       This website's purpose is to help anyone that is interested learn about the basics and history of derivatives. The first thing that might come to mind, is who cares? Valid question. What if knowing and mastering derivatives has a correlation to cancer theory and more applicable things in life. The basics of derivatives and the history have been able to solve many problems throughout history and this website will go through the intriguing background and history that derivatives have. Deriving the product rule and how the formula shows why it works and tells us why the product rule provides us with the derivative of a function, will be a part of the website along with a fascinating connection to find derivatives of a function separate from taking the limit of them.

History and Background:

       Where did derivatives comes from? It is an interesting question, especially from the perspective of being a student and seeing how complicated some formulas and chain rules can become. Makes you wonder who in the world could think about creating this or figuring out the details of it all.

       The first two people that shouldered the creation of many mathematical concepts including derivatives, that are considered to be the fathers of calculus, are Isaac Newton and Gottfried Wilhelm Leibniz. An interesting note about these two mathematicians is that Isaac Newton actually came up with the notation, f ’, to denote a derivative of a function while Leibniz came up with the integral sign of a function while he denoted a derivative of a function y as dy/dx which are all incredibly still used. Especially the dy/dx, this is widely still used to this day in teaching derivatives.

       An interesting story that is given about Newton and Leibniz is that they both seem to have independently created the rules for calculus, but it does not go without a little controversy. Newton from the United Kingdom and Leibniz from Germany both appeared to invent Calculus around the years of 1670. They both say that the other one stole the other persons work, which is fascinating because what are the odds two people both come up with similar research and theories about something so complex as calculus around the same time in two separate countries! It is an interesting argument and theory but they both are pretty widely accepted to have created their work separate from each other. An interesting thing about Isaac Newton was that his motivation to figuring out the rules and laws to calculus was because he did not like to use the infinitesimals in his work.

Explanation of Mathematics:

       Derivatives are simply given in the formula above. The interesting thing about adding h into this formula is that typically the average person is wondering what is happening. It seems beside the point and that we would not get anywhere. The fun thing is that we are actually finding the rate of change in the function. For example, if we get the equation f(x) = x^2 +5x + 6 we can start the process of the mathematics. So, by taking the limit of h while it approaches 0, we add (x+h) in all the spots that originally have x, while then subtracting the original equation making the numerator. Now we have (x+h) ^2 + 5 (x+h) + 6. To simplify the first part of the numerator that is shown the f(x+h) we get x^2 + 2xh + h^2 + 5x +5h +6. Now we have to subtract f(x), in this equation that we are starting with, f(x) = x^2 +5x + 6. So, the final numerator is x^2 + 2xh + h^2 + 5x +5h +6 - (x^2 +5x + 6). An important tip is when you get to this point you look to see what parts of f(x) can subtract from f(x+h). In this scenario and what you find in most situations is you can subtract each part of f(x) from f(x+h). This leaves the numerator as 2xh + h^2 +5h. And to find the derivative, you know need to take the limit with respect to h of the numerator divided by h. So now we have 2xh + h^2 +5h / (h). So, what we can do is divide an h out of the equation which leaves us with 2x +h +5. Then we take the limit as h approaches 0 of the equation 2x + h + 5. This then equals 2x + 5.

       An interesting connection to solving for the derivative of a function by taking the limit of h is, if we graph both the original function and their derivatives by plotting the points, we get some connections. The simplest one is where the original equation of the slope equals 0 you will see that the graph of the derivative will hit the x axis. So, the zeros of the derivative of f are where the slope equals zero for f(x). Another connection for the graphs is where the slope is positive for f(x), the graph of the derivative will be above the x axis, while if the slope is negative it will be below the x axis.

       To prove this, take the original function that we did for the example above, f(x) = x^2 +5x + 6. If we graph it we can see the bottom of the parabola where the slope hits 0 is going to be -2.5. And the slope is negative while it is on the left of -2.5 and the slope is positive when it is on the right of -2.5. So just from those facts from the paragraph above we know that the derivative of the function should cross the x axis at -2.5 and the graph should be above the x axis on the right side of -2.5, while below the x axis on the left side of -2.5. So, taking context clues into consideration, we can look at the line and where the slope is positive and that gives us information on what the derivative is going to look like and where it is above the x axis. Since we know from calculating the derivative above that f’(x) = 2x + 5. Graphing it we see that it crosses the y axis at 5, then can start going down 2 and to the left 1 to figure out where it crosses the x axis. Some points that it will go through are (-1, 3) , (-2, 1), and (-3 , -1). Since the slope is 2 we know it crosses the x axis between the last two, so it crosses the x axis at -2.5. After continuing to graph the derivative we see that it is a line that continues positive to the right of the -2.5 and negative on the left side. This is exactly what we had guessed the derivative of x^2 +5x + 6 would be doing based on what its graph looked like. So, the relationships from the graphs are vital and important to seeing and visualizing how the derivative work and should be graphed itself.

       A mind-blowing connection to finding the derivative of a function instead of using the formula f’(x) = limit of h as it approaches 0 (F (x+h) – F (x) / (h)) there is an amazing connection that works for finding the derivative of a function. If you take the exponent of a variable, multiply it by the leading coefficient of that variable, then minus 1 from the exponent of that variable and do that for every part of the function, you get the derivative. Once again, we will use f(x) = x^2 +5x + 6 as the example. You take the exponent of the variable x^2, which is 2 and multiply it by the leading coefficient of x which is 1 then you get 2, and then the new exponent is just the original exponent, 2, subtract 1 making the new exponent 1. So, we get 2x as the first part. Then if you do it to the second part of the equation, which is 5x then you take the exponent of x which is 1, multiply it by 5 to get 5, then subtract one from the exponent to get 0 as the new exponent. So, we get 5x^0, which we know x^0 = 1, which then gives us 1 multiplied by 5. Now we have 2x+5 which we already know is the derivative, but we can check the last part of 6 for good measures, the exponent of the variable, since there is not one, is 0 so we take 0 and multiply it by 6 and gives us 0 which then multiplied by x^-1 equals 0 thus giving us 2x+5. A very simple system to solving and checking the derivative of a function. Although it is convenient, it still is important to learn the origin of the original equation. Finding the formula and where it comes from is vital to the basic understanding of where derivatives come from as you can visualize what is happening in the original equation.

       Two particularly fundamental formulas that are used in finding derivatives are the product rule and the quotient rule. The math and proof behind the product rule are fascinating and the step by step proof underneath are showed in the pictures (Dawkins). The amazing fact that shows from both of these proofs, and like we talked about earlier, is that all of these functions no matter what or how you solve the derivative the rules of the graphs still apply. When the slope of the orignial function is positive, the graph of the derivative will be above the x-axis. No matter how complex it may seem these rules can stand all the critque and are simply incredible as they prove to be reliable.

Significance and Applicatoins:

       In a lot of different aspects of mathematics or any academic field the common question is, Am I ever going to use this in real life?, which is a valid question. So how are derivatives used in the real world you may ask? Some of the big impacts it is having are in the cancer field. The derivatives have provided research that show some therapy is effective only in animals and have often shown in detail how metabolisms will work. Derivatives have shown that it will help improve the development of cancer therapy and research that only keeps evolving as it still is not perfect (Waxman). That alone is proving to have a vital relevance in how we should view them.

       Some great applications that I see that apply more for students is how to look at the equations and how it can apply to their understanding of what is going on. To see the equation F'(x) = limit of h as it approaches 0 (F (x+h) – F (x) / (h)) and really understand what is happening can be confusing. Bringing it back to the basics is really important to help students visualize what is happening. To do this we will say you have a basic parabola you can take a point, that we will call x, and draw any point along the graph, and draw a straight line between the two points and call it a secant line. Then we can call the distance between point x and the new point h, so we can call it (x+h). So, now lets take these two points become coordinates. Which they will be (x, f(x)) and (x+h, f(x+h)). Now since we have two coordinates we can have use the slope formula which students will love to see at this point. Since slope = y2-y1 all over x2-x1. We get f(x+h) – f(x) / (x+h) -x. So, you can minus x from x from the denominator and which gives you the slope to be f(x+h)-f(x) / h.

       So, after all of that the slope is the same formula that you get to find the derivative of a function and seeing the connection can often be a light bulb moment how these things all work together. Having this application and knowing that the derivative is basically the slope can help understand the graphical side of things which can be a game changer for students. Just like we talked about earlier as well my favorite application through all of this is being able to go between functions and their derivatives and see how the graphs change and represent themselves. One interesting activity that feels like it shows if students understand the graphical representation of derivatives is where you have a handful of function graphs and then have the derivatives all on the other side titled A-L. Then have them recognize the derivatives of the functions and match with the original function. The application of seeing where the slopes equal zero, how the slopes are moving and interpreting that into comprehension is vital to understanding how derivatives work.

Works Cited

Dawkins, Paul. "Theorem, from Definition of Derivative." Calculus I - Proof of Various Derivative Properties,
       https://tutorial.math.lamar.edu/classes/calci/DerivativeProofs.aspx.

Haciomeroglu, E. S., Aspinwall, L., & Presmeg, N. (2009, September). The role of reversibility in the learning of the
       calculus derivative and antiderivative graphs. In Psychology of Mathematics Education (p. 80).

Kovic, J. (2012). The arithmetic derivative and antiderivative. J. Integer Seq, 15.

Kummer, S., & Pauletto, C. (2012, May). The history of derivatives: A few milestones. In EFTA Seminar on Regulation
       of Derivatives Markets (pp. 431-466).

Michell, R. H. (2011). Inositol and its derivatives: their evolution and functions. Advances in enzyme regulation, 51(1),
        84-90.

Ornelas, C. (2011). Application of ferrocene and its derivatives in cancer research. New Journal of Chemistry, 35(10),
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Reissig, H. U., & Zimmer, R. (2003). Donor− acceptor-substituted cyclopropane derivatives and their application
       in organic synthesis. Chemical reviews, 103(4), 1151-1196.

Serhan, D. (2006). The effect of graphing calculators use on students’ understanding of the derivative at a point.
       International Journal for Mathematics Teaching and Learning, (08-05), 30.

Waxman, S., & Anderson, K. C. (2001). History of the development of arsenic derivatives in cancer therapy. The
       oncologist, 6, 3-10.