Partial derivatives and chain rule?

[mohamed el teir]

F(r,s,t,v) = r^2 + sv + t^3, where: r = x^2 +y^2+z^2 /// s = xyz /// v = xe^y /// t = yz^2
find Fxx
i have 2 solutions for this and i am not sure what is the right one
the first solution finds Fx then uses formula : Fxx = Fxr.Rx + Fxs.Sx + Fxv.Vx+Fxt.Tx
the 2nd solution find Fx then uses the formula : Fxx = Frx.Rx + Fr.Rxx + Fsx.Sx + Fs.Sxx + Fvx.Vx + Fv.Vxx = Frr.(Rx)^2+Fr.Rxx + Fss.(Sx)^2 + Fs.Sxx + Fvv.(Vx)^2 + Fv.Vxx

 

[PeroK]

F(r,s,t,v) = r^2 + sv + t^3, where: r = x^2 +y^2+z^2 /// s = xyz /// v = xe^y /// t = yz^2
find Fxx
i have 2 solutions for this and i am not sure what is the right one
the first solution finds Fx then uses formula : Fxx = Fxr.Rx + Fxs.Sx + Fxv.Vx+Fxt.Tx
the 2nd solution find Fx then uses the formula : Fxx = Frx.Rx + Fr.Rxx + Fsx.Sx + Fs.Sxx + Fvx.Vx + Fv.Vxx = Frr.(Rx)^2+Fr.Rxx + Fss.(Sx)^2 + Fs.Sxx + Fvv.(Vx)^2 + Fv.Vxx

It should be easy to figure out which one is correct. Just express $F$ in terms of $x, y, z$ and $t$ and differentiate.

 

[mohamed el teir]

It should be easy to figure out which one is correct. Just express $F$ in terms of $x, y, z$ and $t$ and differentiate.

it was giving me a third answer at first but i discovered a mistake which i made in the calculations, i know the correct one now