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We're given that for $e \in \mathbb{R}^2$ the directional derivative of $u$ in the direction of $e$ is, $$\frac{\partial u}{\partial e}(x,t):= \lim_{h \to 0}\frac{u((x,t) + he) - u(x,t)}{h} = \frac{d}{dh}u((x,t) + he)|_{h=0}$$ and don't understand how they managed to jump from the 'middle' to the 'last' equation in this directional derivative definition. Could someone break this down for me? Cheers!

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up vote 1 down vote accepted

This is basically just definition of the derivative, nothing more.

If you have $$f(h)=u((x,t)+he)$$ then the derivative $f'(h)$ is, by definition, $$f'(0)=\frac{\mathrm{d}f(h)}{\mathrm{d}h}|_{h=0}=\lim\limits_{h\to 0} \frac{f(h)-f(0)}{h-0}=\lim\limits_{h\to 0} \frac{f(h)-f(0)}{h}=\lim\limits_{h\to 0} \frac{u((x,t)+he)-u(x,t)}{h}$$

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