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

Solution

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

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

Note that the $x,y,z$ variables are all expressed in terms of $t$. We can now define the function $G$ as:

$G\left(t\right)=\left(6t+7,\sin\left(2t\right),t^2\right)$

The differential of $G$ is given by the matrix:

$D_G=\begin{bmatrix} 6 \\ 2\cos\left(2t\right) \\ 2t\end{bmatrix}$

We now express the terms in $\nabla F$ over $t$ and multiply $\nabla F$ with $D_G$. This will verify the chain rule.

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

Multiplying $\nabla F$ and $D_G$ will yield the result.

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