The interval \(\displaystyle [a, b]\) is divided into \(\displaystyle n\) equal parts. The devided points are defined as \(\displaystyle \boldsymbol{x}=(x_{0}, x_{1}, x_{2}, \cdots, x_{n})\), where \(\displaystyle x_{0} = a, x_{n} = b\). Let us consider the Riemann integrable \(\displaystyle F(x)\).
In the \(\displaystyle i\)th interval, the mean value theorem derived by Rolle’s theorem, which is based on the axiom of continuity, gives the following equation:
\(\displaystyle \dfrac{F(x_{i+1}) – F(x_{i})}{x_{i+1} – x_{i}} = \left.\dfrac{d F(x)}{d x}\right|_{x=x_{k_{i}}} = F'(x_{k_{i}}) \),
where there is \(\displaystyle x_{k_{i}} \in [x_{i}, x_{i+1}]\). The following equation is defined as follows:
\(\displaystyle f(x_{k_{i}}) = F'(x_{k_{i}}) \).
Because of the Riemann integrable \(\displaystyle F(x)\), the sum of \(F(x)\) in \(\displaystyle n\) equal parts is described as
\(\displaystyle F(b) – F(a) = F(x_{n}) – F(x_{0}) =\sum_{i=0}^{n-1} \left(F(x_{i+1}) – F(x_{i})\right) = \sum_{i=0}^{n-1}f(x_{k_{i}})\left( x_{i+1} – x_{i} \right)\)
\(\displaystyle \rightarrow F(b) – F(a) = \lim_{n\to\infty} F(b) – F(a) = \lim_{n\to\infty} \sum_{i=0}^{n-1}f(x_{k_{i}})\left( x_{i+1} – x_{i} \right) = \int_{a}^{b}f(x)dx \)
\(\displaystyle = \int_{a}^{b}f(t)dt = \int_{a}^{b}\left(\left.\dfrac{d }{dx}F(x)\right|_{x=t}\right)dt \).
By changing the notation of this integral, we get
\(\displaystyle F(x) – F(c) = \int_{c}^{x} f(t) dt \)
\(\displaystyle \rightarrow \dfrac{d}{dx}\left(F(x) – F(c)\right) = \dfrac{d}{dx}F(x) = f(x) = \dfrac{d}{dx}\int_{c}^{x} f(t) dt \)
\(\displaystyle = \dfrac{d}{dx}\int_{c}^{x}\left(\left.\dfrac{d }{dx}F(x)\right|_{x=t}\right)dt \).
Therefore, from the above equation, the fundamental theorem of calculus is as follows:
\(\displaystyle \dfrac{d}{dx}\int_{c}^{x} f(t) dt = f(x) \).
Here, note that since \(\displaystyle F(c)\) is constant, its derivation is zero.
区間\(\displaystyle [a, b]\)を\(\displaystyle n\)等分割する．その分割点は\(\displaystyle \boldsymbol{x}=(x_{0}, x_{1}, x_{2}, \cdots, x_{n})\)と表される．ここで，\(\displaystyle x_{0} = a, x_{n} = b\)であることに注意する．リーマン積分可能な\(\displaystyle F(x)\)を考える．
各区間\(\displaystyle [x_{i}, x_{i+1}]\)毎に，連続の公理を基礎に持つロルの定理から導く平均値の定理より，
\(\displaystyle \dfrac{F(x_{i+1}) – F(x_{i})}{x_{i+1} – x_{i}} = \left.\dfrac{d F(x)}{d x}\right|_{x=x_{k_{i}}} = F'(x_{k_{i}}) \)
となり，\(\displaystyle x_{k_{i}} \in [x_{i}, x_{i+1}]\)となる\(\displaystyle x_{k_{i}}\)が存在する．また，
\(\displaystyle f(x_{k_{i}}) = F'(x_{k_{i}}) \)
と定義する．\(\displaystyle n\)等分割された区間\(\displaystyle [a, b]\)に関する和は，\(\displaystyle F(x)\)がリーマン積分可能であることを用いると，
\(\displaystyle F(b) – F(a) = F(x_{n}) – F(x_{0}) =\sum_{i=0}^{n-1} \left(F(x_{i+1}) – F(x_{i})\right) = \sum_{i=0}^{n-1}f(x_{k_{i}})\left( x_{i+1} – x_{i} \right)\)
\(\displaystyle \rightarrow F(b) – F(a) = \lim_{n\to\infty} F(b) – F(a) = \lim_{n\to\infty} \sum_{i=0}^{n-1}f(x_{k_{i}})\left( x_{i+1} – x_{i} \right) = \int_{a}^{b}f(x)dx \)
\(\displaystyle = \int_{a}^{b}f(t)dt = \int_{a}^{b}\left(\left.\dfrac{d }{dx}F(x)\right|_{x=t}\right)dt \)
となる．この積分の表記を変更すると，
\(\displaystyle F(x) – F(c) = \int_{c}^{x} f(t) dt \)
\(\displaystyle \rightarrow \dfrac{d}{dx}\left(F(x) – F(c)\right) = \dfrac{d}{dx}F(x) = f(x) = \dfrac{d}{dx}\int_{c}^{x} f(t) dt = \dfrac{d}{dx}\int_{c}^{x}\left(\left.\dfrac{d }{dx}F(x)\right|_{x=t}\right)dt\)
となり，微分積分学の基本定理である
\(\displaystyle \dfrac{d}{dx}\int_{c}^{x} f(t) dt = f(x) \)
が成り立つ．ここで，\(\displaystyle F(c)\)は定数であるため，その微分はゼロとなることに注意する．
