# Thread: Power of a function rule v.s. chain rule

1. ## Power of a function rule v.s. chain rule

Hi,

I'm hoping that someone can provide some insight on the following definition:

I'm wanting to understand how this rule is really any different from the chain rule. According to my textbook, the power of a function rule is a special case of the chain rule, but I don't particularly understand what is so different from other cases in which you use the chain rule. Can someone help?

Sincerely,
Olivia

2. ## Re: Power of a function rule v.s. chain rule

Same ... same

Both are the chain rule for composite functions.

$u = g(x)$, just different notation.

3. ## Re: Power of a function rule v.s. chain rule

alright thanks

4. ## Re: Power of a function rule v.s. chain rule

Originally Posted by otownsend
Hi,

I'm hoping that someone can provide some insight on the following definition:

I'm wanting to understand how this rule is really any different from the chain rule. According to my textbook, the power of a function rule is a special case of the chain rule, but I don't particularly understand what is so different from other cases in which you use the chain rule. Can someone help?
If $f(x)=\sin^2(x^3+1) ~\&~g(x)=\sin[(x^3+1)^2]$ then are those two functions different?
If they are the same the we do not have two functions.
Let's evaluate $f\left(\dfrac{3\pi}{5}\right)~\&~g\left(\dfrac{3\ pi}{5}\right)$.
So for simplicity lets say that $\theta= \dfrac{3\pi}{5}$.
Now if you think that $f(\theta)=g(\theta)$ please just stop at once. You most go back and learn basics.
I am sure that you see that they are indeed difference. So how do you a calculator to aid in solving?

In one of those, we cube $\theta$ then add one. Next square that result; finally finding the sine value of that result.
In the other one we cube $\theta$ then add one; square that result & finally find the sine value of that number.is
Now which one is which?

If one does not understand function composition then the chain rule will forever remain a mystery.

Say $u(x)=3\cos^5(x^2+6x)$ then $u'(x)=15\cos^4(x^2+6x)(-\sin(x^2+6x)(2x+6))$
Think how we might evaluate this & reverse the order.