# Find the directional derivative of f at the given point

Find the directional derivative of f at the given point in the direction indicated by the angle theta.
$$\displaystyle{f{{\left({x},{y}\right)}}}={y}{e}^{{-{x}}}$$, $$\displaystyle{\left({0},{4}\right)}$$, $$\displaystyle\theta={\frac{{{2}\pi}}{{{3}}}}$$

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Lindsey Gamble
$$\displaystyle{f{{\left({x},{y}\right)}}}={y}{e}^{{-{x}}}\Rightarrow{{f}_{{x}}{\left({x},{y}\right)}}=-{y}{e}^{{-{x}}}$$ and $$\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}={e}^{{-{x}}}$$. If u is a unit vector in the direction $$\displaystyle\theta={\frac{{{2}\pi}}{{{3}}}}$$, then
$$\displaystyle{D}_{{u}}{f{{\left({0},{4}\right)}}}={{f}_{{x}}{\left({0},{4}\right)}}{\cos{{\left({\frac{{{2}\pi}}{{{3}}}}\right)}}}+{{f}_{{y}}{\left({0},{4}\right)}}{\sin{{\left({\frac{{{2}\pi}}{{{3}}}}\right)}}}=-{4}\times{\frac{{-{1}}}{{{2}}}}+{1}\times{\frac{{\sqrt{{{3}}}}}{{{2}}}}={2}+{\frac{{\sqrt{{{3}}}}}{{{2}}}}$$