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[Step-by-Step] Use the definition to calculate the directional derivatives D_v f(a) for the following functions, if possible: f: R^3 \rightarrow R given by

Question: Use the definition to calculate the directional derivatives $D_{\mathrm{v}} f($ a) for the following functions, if possible:

$f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ given by $f(x, y, z)=x^{2}+y^{2}+z^{2}$ at $\mathbf{a}=(1,1,1)$ in the direction $\mathbf{v}=\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$
$f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ given by $f(x, y)=(x-1)^{2}-y^{2}$ at $\mathbf{a}=(0,1)$ in all directions $\mathbf{v}$;
$f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ given by

\[f(x, y)=\left\{\begin{array}{cc} \frac{2 x y}{\sqrt{x^{2}+y^{2}}}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0) \end{array}\right.\]

at $\mathbf{a}=(0,0)$ in all directions $\mathbf{v}$.

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Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document

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[Step-by-Step] Use the definition to calculate the directional derivatives D_v f(a) for the following functions, if possible: f: R^3 \rightarrow R given by

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