Calculating Derivatives for Dummy Math Students

dy/dx is the instantaneous slope of f(x) at x,y

Popular culture likes to depict mathematically-inclined folks as being weird – Sylvester in "Scorpion," Charlie in "Numb3rs," John Nash in A Beautiful Mind – which we think only goes to show that the people who control popular culture are scared of mathematics. Well, except perhaps Natalie Portman or Danica McKellar, who both have math degrees... Whatever the case, many people avoid math classes like the plague while getting a higher degree. Take Michael O. Smathers of eHow.com, a college history major when he wrote for the site. For Smathers, "How much is 2016-1776?" was about the extent of his mathematical needs; which is why we found it curious that he attempted to explain "How to Calculate Dy/Dx"¹ [please note: the capitalization of "dy/dx" is eHow's].

When our researcher ran across this question, her first thought was, "OK, how much calculus do I remember?" That's because dy/dx is the instantaneous slope of the nonlinear function y = f(x) at a given point, i.e., the slope of a tangent to the curve at that value of x. In other words, it's the first derivative. Which is why we were amazed to read through Michael O's content and find none of the important words nonlinear, curve, function, derivative, tangent, or even calculus anywhere in its depths. Instead Smathers informed his readers that

"Linear algebra involves equations with two variables: x and y. These two variables correspond to a pair of values whose sum is equal to a single constant. Changing the value of either x or y requires the value of the other variable to change to keep the equation true. A line on a coordinate graph represents pairs of x and y values that satisfy the equation. Calculate the ratio of Dy to Dx, or the change in y to the change in x, to predict any value of a linear equation."

It looks like Smathers is getting to the slope... of a linear equation. But that's not what dy/dx is -- he's conflated the derivative with the slope of a linear function, ΔY divided by ΔX, i.e., rise over run. So much for being helpful! Of course, anyone who thinks that a linear equation means that "Changing the value of either x or y requires the value of the other variable to change to keep the equation true..." probably hadn't taken a math course since middle school (although in Smathers' defense, such innumerate wording could be the fault of the content editor, who was most likely an unemployed recent J-school graduate).

Smathers goes on to explain how to calculate the slope of a linear function, either graphically or by placing the function in standard form. In the second, his instructions become a little hazy, but someone who already knows how to perform the operations can follow them (which, we suggest, isn't the point).

When all is said and done, however, all Smathers did was repeat simplistic instructions for calculating the slope of a linear equation -- and that was not the point of the OQ. For lack of space, we'll simply refer the reader to any of several references that do answer the question, from sources like Columbia University or Math Is Fun. Unlike Smathers, neither discusses linear equations... but then neither was written by a Dumbass of the Day!

¹ The original has been deleted by Leaf Group, but can still be accessed using the Wayback machine at archive.org. Its URL was    ehow.com/how_7646553_calculate-dydx.html

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