Trig Derivatives and the CHAIN Rule

Well this is another scribe post by me benofschool, otherwise known as BenchmEn (I'm a plural Justus :P ). Anyways, Dr. E started off the class by introducing us with the derivative of sin(x) and cos(x) our good ol' fundamental trig identities. With the use of our trusty calcumuhlators, we saw that the derivative of sin(x) was cos(x) and cos(x) was -sin(x). What we did was plug the original function in Y1 and in Y2 we inserted the calculator program nDeriv(Y1, X, X). Now we added these two useful derivatives in our handy-dandy derivatives handbook.

The next derivative rule we looked at will be a useful tool in our handbook. It is known as the Chain Rule. Here is the rule (I'll explain soon...) :

d/dx [f(g(x))] = f ' (g(x)) * g ' (x)

Okay so to get this derivative, we first take the derivative of f(x) using keeping g(x) within the f function. We then multiply that result with by the derivative of g(x). Remember the chain rule may involve many layers of functions within functions. So the chain rule may involve the use of your whole arsenal of tools from your handy-dandy derivatives handbook. Also, the chain rule is derived from a number of different derivative rules.

Lets see the chain rule in use...

d/dx [sqrt(x³ + x)]

(x³ + x)^1/2

What we first do is put the function in a way that we know how to work with, with exponents. Now lets take the derivative.

1/2(x³ + x)^-1/2 * 3x² + 1

So what I did here was to use the power rule on the original function. The exponent of the original function becomes the coefficient and the new exponent will be the exponent of the original function minus 1. The power rule requires to multiply that new function by the derivative of the function within the brackets. The function in the brackets is simply a polynomial and we've done those tons of time before. Power rule galore. Now lets finish it off

3x² + 1
----------------
2(x³ + x)^1/2

Since the polynomial in the brackets had a negative exponent we have to take the reciprocal of it or throw it into the denominator. Now it gives us the beast above. Algebra is such a beautiful thing. But remember the first line was quite ugly as well. So all we needed to do was algebraically massage the beast using our massaging tools that we have collected over the years.

That was our calc hour, homework is chapter 4.3 all odd numbers I believe (Kristina *ahem*) Speaking of Kristina!!! She won't be our next scribe, but Not Paul.

Good Night and remember Manipulation is a beautiful thing....

Pants are essential