Chain Rule for Quotients: Explained!

[ziddy83]

Hey what's up,
I had a question on the chain rule...How would I use the chain rule on a quotient...like if i have 1/(t^4 + 1)^3 , Would I use the quotient rule first, or just start with the chain rule?

 

[mspaic]

ok use the chain rule SO:

(bottom*d of top - top (d of bottom) )/bottom squared


wen u do all that u get (i may be wrong though)

-12t^3/(t^4 +1)^4

 

[mspaic]

there are many rules u can use

like i used the quotient rule there
u can also solve it by using the prduct rule (WHICH I GOT WRONG ON THE TEST ERRR)

 

[ziddy83]

cool..thanks man.

 

[Justin Lazear]

Either one works.

Quotient rule:

$$f(t) = \frac{p(t)}{q(t)} = \frac{1}{(t^4 + 1)^3}$$
so
$$p(t) = 1$$
and
$$q(t) = (t^4+1)^3$$
which are both functions of t.

Alternatively, the chain rule:

$$f(t) = f(u(t)) = \frac{1}{u^3}$$
where $u(t) = t^4 + 1$

So we have

$$\frac{d}{dt}f(u(t)) = \frac{df}{du}\frac{du}{dt}$$
$$= \frac{d}{du}\left( u^{-3} \right) \frac{d}{dt}\left( t^4 + 1 \right) = (-3u^{-4})\cdot (4t^3) = \frac{-12t^3}{(t^4+1)^4}$$

I imagine the chain rule method is a bit faster, and I personally think I'd be more likely to make a silly mistake with the quotient rule, so.

--Justin