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!
The definition of a directional derivative
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derivatives
1 Answers
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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, $f'(0)=\frac{\mathrm{d}f(h)}{\mathrm{d}h}|_{h=0}=\lim\limits_{h\to 0} \frac{f(h)-f(0)}{h-0}=\lim\limits_{h\to 0} \frac{f(h)-f(0)}{h}=\lim\limits_{h\to 0} \frac{u((x,t)+he)-u(x,t)}{h}$