Deriving with multiple variables

In summary, the conversation discusses a problem with deriving the equation for relativistic energy and the use of the chain rule to solve it. The student is stuck on a specific step and has tried various techniques to solve it. The solution is to use the chain rule and substitute for the relationship between speed and time.
  • #1
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Homework Statement


I'm stuck at one of the derivations for relativistic energy. I've figured out every other single step, but I just can't wrap my head around this one:

Prove that:

[tex] {\frac{d}{dt}} {\frac {mu} {\sqrt{1-u^2/c^2}}} = {\frac {m {\frac{du} {dt}}} {(1-u^2/c^2)^{3/2}}} [/tex]


Homework Equations


u is speed, so:

[tex] u = dx/dt [/tex]
I don't know if that's helpful.


The Attempt at a Solution


I've tried everything that's in my calculus toolbox, but I guess that's not a whole lot. I know how to derive basic functions, but I just can't seem to figure out how turn this into something I can work with. It says to derive to t, but there's not even a t in the function. I know that u and t are related in a way, but substituting just leads to more trouble.
 
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  • #2
Speed depends on time, u = u(t), so you just need to use the chain rule:[tex] \frac{d f(u(t))}{dt} = \frac{df}{du} \frac{du}{dt} [/tex]
 
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  • #3
Alright, I've got it now. Thanks!
 

Related to Deriving with multiple variables

1. What is the purpose of deriving with multiple variables?

Deriving with multiple variables allows us to find the rate of change of a function with respect to more than one variable. This can help us understand the relationship between multiple variables and how changes in one variable affect the other.

2. How do you take the derivative of a function with multiple variables?

To take the derivative of a function with multiple variables, we use the partial derivative notation (∂) and take the derivative with respect to one variable while holding the others constant. This results in a partial derivative for each variable in the function.

3. What is the difference between a partial derivative and a total derivative?

A partial derivative is taken with respect to one variable while holding the others constant, while a total derivative is taken with respect to all variables in the function. In other words, a total derivative takes into account the relationship between all variables, while a partial derivative only looks at the relationship with one variable.

4. Why is it important to consider multiple variables when taking a derivative?

Considering multiple variables allows us to better understand the relationship between different quantities in a function. This can be useful in fields such as physics, economics, and engineering, where multiple variables often interact and affect each other.

5. What are some common applications of deriving with multiple variables?

Deriving with multiple variables has many real-world applications, such as finding the maximum or minimum value of a function, optimizing processes, and understanding the relationship between variables in a system. It is also used in fields like finance, computer science, and statistics to analyze complex data and make predictions.

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