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Chain Rules

  • Thread starter nothing123
  • Start date
  • #1
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It's been a while since I've taken calculus. I was going through the derivation of the work-energy theorem and came across this: dv/dt = (dv/dx)(dx/dt) which is supposed to be a result of the chain rule. Anyone care to explain and please simplify it as much as possible.

Thanks!
 

Answers and Replies

  • #2
nicksauce
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The chain rule is

f(g(x))' = f'(g(x))g'(x)

Now replace f with v, g with x, x with t,the first ' with d/dt, the second ' with d/dx (since f is a function of g aka x), and the third ' with d/dt and voila you have
dv/dt = (dv/dx)(dx/dt)
 
  • #3
tiny-tim
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Hi nothing123! :smile:

If v is a function of x only, and x is a function of t only, then if you increase t by a small amount ∆t, then x increases by a small amount ∆x = (dx/dt)∆t. (1)

But v also increases, by a small amount ∆v = (dv/dx)∆x. (2)

So, combining (1) and (2):
∆v = (dv/dx)∆x = (dv/dx)(dx/dt)∆t. :smile:
 
  • #4
97
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Great, thanks for your help guys.
 

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