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

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To find the second derivative of multivariable functions using the chain rule, the discussion clarifies the application of derivatives with respect to time variables t. The initial expression for the first derivative, Ft, is confirmed as correct. The revised expression for the second derivative, Ftt, incorporates both mixed partial derivatives Fxy and Fyx, indicating their necessity in the calculation. The conversation emphasizes the importance of applying both the product rule and chain rule accurately in this context. Overall, the approach to deriving the second derivative is validated through collaborative correction.
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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