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If f(x,y,z)=x2y3+xyzf(x, y, z)=x^{2}-y^{3}+x y z, and x=6t+7,y=sin2t,z=t2x = 6t + 7, y = \sin 2 t, z=t^{2}, verify the chain rule by finding df/dt in two different ways.


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We verify the chain rule by using the rule (4) on page 147. First we calculate F\nabla F

F=(2x+yz,3y2+xz,xy)\nabla F=\left(2x+yz,-3y^2+xz,xy\right)

Note that the x,y,zx,y,z variables are all expressed in terms of tt. We can now define the function GG as:


The differential of GG is given by the matrix:

DG=[62cos(2t)2t]D_G=\begin{bmatrix} 6 \\ 2\cos\left(2t\right) \\ 2t\end{bmatrix}

We now express the terms in F\nabla F over tt and multiply F\nabla F with DGD_G. This will verify the chain rule.

F=(2(6t+7)+t2sin(2t),3sin2(2t)+(6t+7)t2,(6t+7)sin(2t))\nabla F=\left(2\left(6t+7\right)+t^2\sin\left(2t\right),-3\sin^2\left(2t\right)+\left(6t+7\right)t^2,\left(6t+7\right)\sin\left(2t\right)\right)

Multiplying F\nabla F and DGD_G will yield the result.

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