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I am taking some time to review differentials. What I don't quite get is why the change along the tangent line is $f'(x) \Delta x$, and how it leads to $f'(x)dx$

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up vote 2 down vote accepted

Indeed, it is the definition of derivative of a function at a point. we know that

$$\frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}$$

for derivative at a point $x$ we have:

$$\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}= y'(x) = m $$

so for each $\Delta x$ that is small enough ,you can approximate $f(x)$ within the interval $[x-\delta x , x +\delta x]$ with a line with a slope of $m=f'(x)$ and write: (approximately)

$$\Delta y = f'(x)\Delta x$$

again, in the limit where ${\Delta x\to 0}$ , we get $dy=f'(x)dx$ .Now there isn't any approximation here and the relation is correct with infinite Accuracy at each point.

you can see "derivative" in wikipedia , where it is a Featured Article.

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