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

*x*², which leads us to believe that an increase in

*y*will result in an increase in

*x*². We represent this by a slight increase (

*d*) to both sides of the equation, giving us

**.**

*y*+*dy*=(*x*+*dx*)²We can then FOIL out the right side to get the equation

**y****+**.

*dy*=*x*²+2*x*(*dx*)+*dx*²Here, Thompson explains that as our last term,

*dx*², 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*=*x*²+2x(dx)We know from the first step that

*y*is equal to

*x*², so we can cancel both those terms out, giving us

**.**

*dy*=2*x*(*dx*)Since we want one side of our equation to be

*dy*/

*dx*, we can divide both sides by

*dx*, giving us

**, which tells us that “[t]he ratio of the growing of**

*dy*/*dx*=2*x**y*to the growing of

*x*is […] 2

*x*” (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!