Calculating total derivative of multivariable function

In summary: You have summarized the conversation about deriving a result from a textbook for a user. The user has provided their solution and is asking for confirmation or any pointers. The solution involves finding the derivative of the current and voltage functions, which are both functions of a single variable, q1. The user has calculated the derivative and is unsure if it is correct. They are also wondering how to incorporate another variable, q2, into the answer.
  • #1
halleffect
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0
Homework Statement
Here is the full problem statement (with equations): https://i.imgur.com/z2SWodC.png The answer that I'm trying to derive is at the bottom.
Relevant Equations
All equations are also included in the above image (https://i.imgur.com/z2SWodC.png).
This isn't a homework problem exactly but my attempt to derive a result given in a textbook for myself. Below is my attempt at a solution, typed up elsewhere with nice formatting so didn't want to redo it all. Direct image link here. Would greatly appreciate if anyone has any pointers.

?hash=c7716cab9784b847841ac513ff5f38b2.png
?hash=c7716cab9784b847841ac513ff5f38b2.png
 

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  • #2
By ##I_0## and ##V_0## sum conditons, we know
[tex]i=2i_1-I_0=i(q_1)[/tex]
[tex]v=2v_1-V_0=v(q_1)[/tex]
They both are functions of variable ##q_1## only.
[tex]\frac{di}{dv}=\frac{\frac{di}{dq_1}} {\frac{dv}{dq_1}}=\frac{\frac{di_1}{dq_1}} {\frac{dv_1}{dq_1}}[/tex]
we can calculate it.
 
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  • #3
anuttarasammyak said:
By ##I_0## and ##V_0## sum conditons, we know
[tex]i=2i_1-I_0=i(q_1)[/tex]
[tex]v=2v_1-V_0=v(q_1)[/tex]
They both are functions of variable ##q_1## only.
[tex]\frac{di}{dv}=\frac{\frac{di}{dq_1}} {\frac{dv}{dq_1}}=\frac{\frac{di_1}{dq_1}} {\frac{dv_1}{dq_1}}[/tex]
we can calculate it.

Thanks very much for the reply; with that info:

[tex]\frac{di_1}{dq_1} = 2q_1+1[/tex]
[tex]\frac{dv_1}{dq_1} = U_T(\frac{2q_1+1}{q_1})[/tex]

And from this I get

[tex]\frac{di}{dv}=\frac{ \frac{di_1}{dq_1} }{ \frac{dv_1}{dq_1} }= \frac{q_1}{U_T}[/tex]

I'm not sure if this is correct. For one thing, I'm not sure how I could incorporate ##q_2## into the answer like the given one (which is ##\frac{2}{U_T}\frac{q_1 q_2}{q_1+q_2}##), although it does seem to make sense that it should be expressible through just one of the ##q##'s. Is there a way to show this to be equal to the given answer? (Assuming I haven't made an error here)
 

Related to Calculating total derivative of multivariable function

1. What is the total derivative of a multivariable function?

The total derivative of a multivariable function is a measure of how the output of the function changes with respect to changes in all of its input variables. It takes into account the effects of all the input variables on the output, rather than just one variable at a time.

2. Why is it important to calculate the total derivative?

Calculating the total derivative allows us to understand how small changes in the input variables affect the output of a function. This is crucial in many fields of science, such as physics and economics, where small changes can have significant impacts.

3. How do you calculate the total derivative of a multivariable function?

The total derivative can be calculated using partial derivatives, which measure the rate of change of the function with respect to each input variable. These partial derivatives are then combined using the chain rule to find the total derivative.

4. What is the difference between the total derivative and the partial derivative?

The partial derivative only measures the rate of change of a function with respect to one input variable, while the total derivative takes into account the effects of all the input variables on the output. In other words, the total derivative gives a more comprehensive understanding of how the function changes with respect to all its inputs.

5. Can the total derivative be negative?

Yes, the total derivative can be negative. This means that as one or more of the input variables increases, the output of the function decreases. It is important to consider both positive and negative values of the total derivative when analyzing the behavior of a multivariable function.

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