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 $f'(x)dx$


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.