Understanding the Relationship between dv/dt and vdv/dx in Calculus

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SUMMARY

The relationship between dv/dt and vdv/dx in calculus is established through the chain rule of differentiation. Specifically, the equation dv/dt = (dv/dx)(dx/dt) illustrates how the rate of change of velocity with respect to time can be expressed in terms of the rate of change of velocity with respect to position. This relationship clarifies the connection between differentials and their respective variables, reinforcing the importance of understanding the chain rule in calculus.

PREREQUISITES
  • Understanding of basic calculus concepts, including derivatives.
  • Familiarity with the chain rule of differentiation.
  • Knowledge of the product rule in calculus.
  • Ability to manipulate differential equations.
NEXT STEPS
  • Study the chain rule of differentiation in depth.
  • Explore the product rule and its applications in calculus.
  • Practice solving differential equations involving velocity and time.
  • Learn about the implications of derivatives in physics, particularly in motion analysis.
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Students studying calculus, educators teaching differentiation concepts, and anyone seeking to deepen their understanding of the relationship between velocity and time in mathematical contexts.

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I have a question that asks me to write dv/dt as a function of x.

I know that dv/dt = vdv/dx

but I don't know why this is. Could anybody shed some light on this for me? Just so I understand why it is this!

Thanks in advance
 
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Did you hear about the chain rule of differentiation...?

Daniel.
 
dextercioby said:
Did you hear about the chain rule of differentiation...?

Daniel.

yeah that though occurred to me whilst sitting in the car just.

but surely there's another term involved in it isn't there?
 
No.

\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}.

Daniel.
 
dextercioby said:
No.

\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}.

Daniel.

Of course! What I had in mind was the product rule! Thank you for that - makes so much sense to me now!
 

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