If $f(x, y)$ is differentiable at $(a, b)$, then $D_u(a, b) = D_v(a, b)$ for all unit vectors $u$ and $v$.
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Let $f(x, y)=(x y)^{1 / 3}.$ \begin{equation} \begin{array}{l}{\text { (a) Use the limit definition to show that } f_{x}(0,0)=f_{y}(0,0)=0.} \\ {\text { (b) Use the limit definition to show that the directional derivative }} \\ {D_{\mathrm{u}} f(0,0) \text { does not exist for any unit vector u other than } \mathrm{i} \text { and } j .} \\ {\text { (c) Is } f \text { differentiable at }(0,0) ?}\end{array} \end{equation}
DIFFERENTIATION IN SEVERAL VARIABLES
The Gradient and Directional Derivatives
If $f(x+y)=f(x)+f(y), \forall x, y \in R$ and $f(x)$ is a differentiable function, find $f(x)$.
The Continuity and Differentiability
Level I
A function $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ is such that $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x}) \cdot \mathrm{f}(\mathrm{y})$ for all $\mathrm{x}, \mathrm{y}$ in $\mathrm{R}$ and $\mathrm{f}(\mathrm{x}) \neq 0$ for any $\mathrm{x}$ in $\mathrm{R}$. If $\mathrm{f}(\mathrm{x})$ is differentiable and $\mathrm{f}^{\prime}(0)=2$, then (a) $\mathrm{f}^{\prime}(\mathrm{x})=2 \mathrm{f}(\mathrm{x})$ (b) $\mathrm{f}(\mathrm{x})=2 \mathrm{f}^{\prime}(\mathrm{x})$ (c) $\mathrm{f}(\mathrm{x})=\mathrm{f}^{1}(\mathrm{x})$ (d) $\mathrm{f}^{\prime}(\mathrm{x})=-\mathrm{f}(\mathrm{x})$
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