Today’s soundtrack is Dethklok: Dethalbum II, a death metal album that is actually a parody album that ties in with the cartoon “Metalocalypse.”
Calculus Made Easy (Silvanus Thompson)
Chapter 4: Simplest Cases
Today, I’ll finally be learning to differentiate a few simple algebraic expressions. As we know from past lessons, our goal is to find the value dy/dx – the “proportion between the growing of y and the growing of x” (p. 51).
The first example given is the classic equation y=x².
We can see straight off that y is equal to x², which leads us to believe that an increase in y will result in an increase in x². We represent this by a slight increase (d) to both sides of the equation, giving us y+dy=(x+dx)².
We can then FOIL out the right side to get the equation y+dy=x²+2x(dx)+dx².
Here, Thompson explains that as our last term, dx², is infinitely small (being “a small quantity of the second order of smallness” (p. 52), we can safely discard it. So we now have y+dy=x²+2x(dx).
We know from the first step that y is equal to x², so we can cancel both those terms out, giving us dy=2x(dx).
Since we want one side of our equation to be dy/dx, we can divide both sides by dx, giving us dy/dx=2x, which tells us that “[t]he ratio of the growing of y to the growing of x is […] 2x” (p. 52)!
Thompson goes through several other examples of differentiating, then finds a pattern in the solutions, bringing us to the power rule, which is often used to differentiate “low-order functions” (p. 57). The rule is this: “To differentiate xⁿ, multiply it by the exponent and reduce the exponent by one, so giving us nxⁿ⁻¹ as the result” (p. 57).
While going through the exercises, I was able to deduce the following: if we are dealing with a root of a x, n is x‘s power over the root’s index.
That’s it for today!