# I can't understand the relation between derivative and integration.

Why (not how for how have been a proof) the integration of derivative of a function give us the function :| let me explain more with pictures.

like here we have the derivative of $$x^2$$,

but how the hell does the area of under it give us the $$f(x)$$

I can't understand the intuition of it or something.

• @AnotherUser: The images are plots. Perhaps OP should have more exposition, but these plots cannot be reduced to anything searchable. Commented Jun 19, 2022 at 17:28
• For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Commented Jun 19, 2022 at 17:29
• OP: Why did you revert my edit to inline your images? Doing so makes your question less legible, not more. Commented Jun 19, 2022 at 17:29
• @BrianTung i couldn't ask it without images Commented Jun 19, 2022 at 17:36
• "i will do it in my next questions thanks". Why don't you do it with this question? It is totally permitted, even expected, that new users will have to make a couple of edits to get their first question up to the standards required. Commented Jun 19, 2022 at 17:46

Consider a differentiable function $$f$$ and let $$x, y$$ be such that $$y=f(x)$$.
The Fundamental Theorem of Calculus states that $$f(b)-f(a)=\int_{a}^{b}f'(x)dx$$ where $$f'$$ is the derivative of $$f$$.
The left hand side represents the difference between two values of $$f$$. The expression $$f'(x)dx$$ represents a small step in the $$y$$ direction, i.e. $$dy=f'(x)dx$$ and the integral sums up those small changes to get the total change.
Therefore, we can translate the above theorem to say that the total change in $$f$$ that occurs when $$x$$ goes from $$a$$ to $$b$$ can be computed by summing all of the small changes that occurred.