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The following function represents a curve in a two-dimensional plane. $$f(x,y)=x\arctan(\frac{x}{y})$$

Determine the value of $\frac{\partial f}{\partial \overrightarrow{u}}(1,1)$, where $u$ points to the direction of maximum growth for the function at the given point.

I'm trying to understand this problem, but I really don't know what I should do to solve it. I would like to know your opinions and possible solutions regarding this exercise. What results should I use to solve it and how should I proceed to find its solution? How can I interpret it geometrically?

Curious
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1 Answers1

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Notice that $f(x,y)=x\arctan(x/y)$ is differentiable function at $(x_{0},y_{0}):=(-1,1)$, then $$\max_{|\|\vec{u}\|=1}\frac{\partial f}{\partial \vec{u}}(x_{0},y_{0})=\|\nabla f(x_{0},y_{0})\|$$ i.e., the maximum of the directional derivative occurs when $\vec{u}$ has the same direction as the gradient vector $\nabla f$.

  • The gradient vector $\nabla f$ can be find as $\nabla f=(f_{x},f_{y})$ and the norm of gradient vector in the plane can be find $\|\nabla f\|=\sqrt{f_{x}^{2}+f_{y}^{2}}$.
A. P.
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