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Is there a way to visualize (like a picture in mind) the $n$-th derivative ?

For $n=1$ is the tangent line and we can visualize it quite well.

More abstractly is it possible to see the geometric significance of the $n$-th derivative for all $n$?

For example can the human eye distinguish the graph of a $C^4$ function from a $C^5$ function ?

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The tangent line is the first order Taylor polynomial. The n:th derivative can be "visualized" in the same sense as the n:th order Taylor polynomial. This will only give a localized significance to the derivative.

There are many other important properties of derivatives. For instance when solving differential equations the exponential functions are very important since they are "their own" derivatives. Then splitting the function into sums of exponentials may help solve the problem.

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