Implicit differentiation and chain rule-

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The discussion centers on using implicit differentiation and the chain rule to find the derivative dr/dq for the function R = e^(q+p), where p is defined by the equation q^2*p + p^2*q + qp = 3. The user questions whether they should substitute the equation for p into R and expresses uncertainty about their derivation process, concluding that dr/dq seems incorrect. They derive that dr/dq equals zero, but they struggle to connect this with the function provided. The user seeks clarification on whether they need to create separate chains for dp/dq and incorporate that into their calculation for dr/dq.
SavvyAA3
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The question and my workings are attached:
 
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Sorry it seems that attachments do not upload:

Here is the Question:

Suppose R = R(q,p) = e^(q+p), where p = p(q) is defined through the equation

q^2*p+p^2*q+qp = 3

Letn r(q) = R(q,p(q)). Use the chain rule to calculate the derivative dr/dq at the point q=1.


Can you please tell me if it is correct to assume that this question is asking you to insert q^2*p+p^2*q+qp = 3 into e^(q+p), for p?

From there I have tried to derive the chain for dr/dq:

My workings:

since e^(q+q^2*p+p^2*q+qp) = 3, a constant

dr/dq = 0

so the chain
(∂R)/(∂q)*dq/(dq )+(∂R)/(∂r)*dr/(dq)= 0

 (∂R)/(∂r)*dr/(dq)= - (∂R)/(∂q)

 dr/(dq)= - (∂R)/(∂q )* (dr)/(dR)

dr/(dq)= - (∂R)/((∂q )/((∂R)/(∂r)))


I don't think this is the correct chain because I can’t find dr/dqfrom the function given.

I would be most grateful if you could point me along the correct direction for this.
Must I create two chains? i.e. one for dp/dq and then somehow incorporate this for dr/dq?

Thanks.
 

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