How to Use Chain Rule to Find Second Derivative of Multivariable Functions?

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Homework Help Overview

The discussion revolves around the application of the chain rule to find the second derivative of multivariable functions, specifically focusing on functions of two variables where each variable is a function of a third variable, t.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to derive the second derivative using the chain rule and are discussing the inclusion of mixed partial derivatives in their expressions. There is a focus on ensuring the correct application of both the product rule and chain rule in their calculations.

Discussion Status

The discussion is active, with participants refining their expressions and confirming the correctness of their formulations. There is acknowledgment of potential errors and corrections made in the notation of derivatives, indicating a collaborative effort to clarify the approach.

Contextual Notes

Participants are working under the constraints of homework rules, which may limit the extent of guidance provided. There is an emphasis on ensuring all relevant terms are included in the derivative expressions.

TranscendArcu
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Suppose I have

F(x,y) and y=y(t) and x=x(t)

Therefore,

Ft = Fx*xt + Fy*yt. Right?

Can I write

Ftt = (Fxx*xt + Fyy*yt)*xt + Fx*xtt + (Fxx*xt + Fyy*yt)*yt + Fy*ytt

?

Basically I'm trying to figure out the second derivative by chain rule.
 
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Shouldn't there be some F_xy's and F_yx's in there?
 
You're right. I meant to write:


Ftt = (Fxx*xt + Fxy*yt)*xt + Fx*xtt + (Fyx*xt + Fyy*yt)*yt + Fy*ytt
 
TranscendArcu said:
You're right. I meant to write:


Ftt = (Fxx*xt + Fxy*yt)*xt + Fx*xtt + (Fyx*xt + Fyy*yt)*yt + Fy*ytt

Looks ok now, good job. Just product rule and chain rule, yes?
 

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