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In the question here, Directional derivative is given as $$D_{\mathbf v}\mathbf g(\mathbf p) := \lim_{\theta\to 0}\frac{\mathbf g(\mathbf p+\theta\mathbf v)-\mathbf g(\mathbf p)}{\theta} \\ \\ \text{Here v is a unit vector}$$ - which is the same one I studied in high school.

However, I came across this Gateaux Derivative, $$ \begin{align*} \lim_{ \theta \to 0} \frac{F[f+\theta h]- F[f]}{\theta} \end{align*} \\ \\ \text{Here h is just a vector}$$

These definitions are so similar. I really cannot appreciate the difference between them. Can someone help me understand this with some simple examples? Thank you.

P.S: I am just a beginner and I am not familiar with Banach Spaces and stuff.

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They are basically the same thing. Gateaux is between any two locally convex topological vector spaces. Directional derivative is just for $f:\Bbb R^n\to \Bbb R$.

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