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Chain Rule Question

  1. Nov 4, 2004 #1
    Hey whats 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?
     
  2. jcsd
  3. Nov 5, 2004 #2
    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
     
  4. Nov 5, 2004 #3
    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)
     
  5. Nov 5, 2004 #4
    cool..thanks man.
     
  6. Nov 5, 2004 #5
    Either one works.

    Quotient rule:

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

    Alternatively, the chain rule:

    [tex]f(t) = f(u(t)) = \frac{1}{u^3}[/tex]
    where [itex]u(t) = t^4 + 1[/itex]

    So we have

    [tex]\frac{d}{dt}f(u(t)) = \frac{df}{du}\frac{du}{dt}[/tex]
    [tex] = \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}[/tex]

    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
     
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