The fundamental theorem of differential and integral calculus is as follows:
\(\displaystyle \dfrac{d}{dx} \int_{c}^{x} f(t)dt = f(x) \),
where \(t\) is a dummy variable, \(x\) is a variable or point of interest, and \(c\) is a conveniently selected and fixed point.
It is not strictly defined that how widely the difference should be set or what kind of segmentation should be used to calculate the sum, but if it can be chosen conveniently (see Mathematics for details), then the left-hand side is
\(\displaystyle \dfrac{d}{dx} \int_{c}^{x} f(t)dt \approx \dfrac{1}{\Delta t}\left(\sum_{c}^{x+\Delta t} f(t) \Delta t- \sum_{c}^{x} f(t’) \Delta t’\right) \approx \dfrac{1}{\Delta t}\left(\sum_{c}^{x+\Delta t} f(t) \Delta t- \sum_{c}^{x} f(t) \Delta t\right)\)
\(= \dfrac{1}{\Delta t}f(x+\Delta t)\Delta t = f(x+\Delta t) \approx f(x) \).
The above result derives the right-hand side of the fundamental theorem of differential and integral calculus (we use the infinitesimal quantity \(\Delta t\) etc.).
Geometrically, the left side has the height \(f(t)\) and width \(\Delta t\). The difference between the two sums is the small area \(f(x)\Delta t\) formed by \(f(x)\) and \(\Delta t\). In other words, the left side indicates the height \(f(x)\) of a small area \(f(x)\Delta t\) with the slope (differentiation) of the function (integral).
微分積分学の基本定理は下記の通りである（\(t\)はダミー変数で，\(x\)は変数や注目点であり，\(c\)は都合よく選択された定点である）：
\(\displaystyle \dfrac{d}{dx} \int_{c}^{x} f(t)dt = f(x) \).
どのくらいの幅で差分化したり，どのような区分化で和を取るのかは，厳密に定義されていないが、都合よく（詳細は数学を参照）選ぶことができるとすれば，その左辺は
\(\displaystyle \dfrac{d}{dx} \int_{c}^{x} f(t)dt \approx \dfrac{1}{\Delta t}\left(\sum_{c}^{x+\Delta t} f(t) \Delta t- \sum_{c}^{x} f(t’) \Delta t’\right) \approx \dfrac{1}{\Delta t}\left(\sum_{c}^{x+\Delta t} f(t) \Delta t- \sum_{c}^{x} f(t) \Delta t\right)\)
\(= \dfrac{1}{\Delta t}f(x+\Delta t)\Delta t = f(x+\Delta t) \approx f(x) \)
となり，微分積分学の基本定理の右辺が導かれる（微小量\(\Delta t\)などを用いた）．
幾何学的には，左辺は高さ\(f(t)\)と横幅\(\Delta t\)が成す微小面積を足し上げたものの差分は微小面積\(f(x)\Delta t\)そのものを意味している．つまり，左辺は関数（積分）の傾き（微分）により，微小面積\(f(x)\Delta t\)の高さ\(f(x)\)が求められることを幾何学的には表している．
