微積分学 (calculus)

$$
f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}
$$

f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}

$$
\frac{d}{dx} \{ \alpha f(x) + \beta g(x) \} =\alpha f'(x) + \beta g'(x)
$$

\frac{d}{dx} \{ \alpha f(x) + \beta g(x) \} =\alpha f'(x) + \beta g'(x)

$$
\frac{d}{dx} (f(x) g(x)) = f'(x) g(x) + f(x) g'(x)
$$

\frac{d}{dx} (f(x) g(x)) = f'(x) g(x) + f(x) g'(x)

$$
\frac{d}{dx} \{ f(x)^n \} = nf(x)^{n-1} f'(x)
$$

\frac{d}{dx} \{ f(x)^n \} = nf(x)^{n-1} f'(x)

$$
\frac{d}{dx} \left(\frac{f(x)}{g(x)} \right) = \frac{f'(x) g(x) – f(x) g'(x)}{g(x)^2}
$$

\frac{d}{dx} \left(\frac{f(x)}{g(x)} \right) = \frac{f'(x) g(x) – f(x) g'(x)}{g(x)^2}

\begin{eqnarray*}
f'(x) &=& \lim_{h \to 0} \frac{\sin(x+h) – \sin x}{h}\\
\\
&=& \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h – \sin x}{h}\\
\\
&=& \lim_{h \to 0} \frac{\cos x \sin h}{h} \\
\\
& & \sin h \le h \le \tan h \ であり、 \ 1 \ge \frac{\sin h}{h} \ge \cos h より
\\
\\f'(x) &=& \lim_{h \to 0} \cos x \cdot \frac{\sin h}{h} \\
\\
&=& \cos x
\end{eqnarray*}

f'(x) &=& \lim_{h \to 0} \frac{\sin(x+h) – \sin x}{h}

&=& \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h – \sin x}{h}

&=& \lim_{h \to 0} \frac{\cos x \sin h}{h}

& & \sin h \le h \le \tan h \ であり、 \ 1 \ge \frac{\sin h}{h} \ge \cos h より

f'(x) &=& \lim_{h \to 0} \cos x \cdot \frac{\sin h}{h}

&=& \cos x

\begin{eqnarray*}
f'(x) &=& \lim_{h \to 0} \frac{\cos(x+h) – \cos x}{h}\\
\\
&=& \lim_{h \to 0} \frac{\cos x \cos h – \sin x \sin h – \cos x}{h}\\
\\
&=& – \lim_{h \to 0} \frac{\sin x \sin h}{h} \\
\\
& & \sin h \le h \le \tan h \ であり、 \ 1 \ge \frac{\sin h}{h} \ge \cos h より
\\
\\f'(x) &=& – \lim_{h \to 0} \sin x \cdot \frac{\sin h}{h} \\
\\
&=& -\sin x
\end{eqnarray*}

f'(x) &=& \lim_{h \to 0} \frac{\cos(x+h) – \cos x}{h}

&=& \lim_{h \to 0} \frac{\cos x \cos h – \sin x \sin h – \cos x}{h}

&=& – \lim_{h \to 0} \frac{\sin x \sin h}{h}

& & \sin h \le h \le \tan h \ であり、 \ 1 \ge \frac{\sin h}{h} \ge \cos h より

f'(x) &=& – \lim_{h \to 0} \sin x \cdot \frac{\sin h}{h}

&=& -\sin x

\begin{eqnarray*}
y &=& \log _a x \\
\\
e &=& 2.7182818 \cdots\\
\\
e &=& \lim _{t \to \infty} \left( 1 + \frac{1}{t} \right)^t
\end{eqnarray*}

y &=& \log _a x

e &=& 2.7182818 \cdots

e &=& \lim _{t \to \infty} \left( 1 + \frac{1}{t} \right)^t

\begin{eqnarray*}
\frac{df(x)}{dx} &=& \lim _{h \to 0} \frac{e^{x+h} – e^x}{h} \\
\\
&=& \lim _{h \to 0} \frac{e^x (e^h -1)}{h} \\
\\
&=& e^x
\end{eqnarray*}

\frac{df(x)}{dx} &=& \lim _{h \to 0} \frac{e^{x+h} – e^x}{h}

&=& \lim _{h \to 0} \frac{e^x (e^h -1)}{h}

&=& e^x

\begin{eqnarray*}
\lim _{h \to 0} \frac{e^h – 1}{h} &=& \lim _{t \to \infty} \frac{1}{\ln \left( 1+ \frac{1}{t} \right)^t} \\
\\
&=& \frac{1}{\ln e} \\
\\
&=& 1
\end{eqnarray*}

\lim _{h \to 0} \frac{e^h – 1}{h} &=& \lim _{t \to \infty} \frac{1}{\ln \left( 1+ \frac{1}{t} \right)^t}

&=& \frac{1}{\ln e}

&=& 1

\begin{eqnarray*}
\frac{df(x)}{dx} &=& \lim _{h \to 0} \frac{\ln (x+h) – \ln x}{h} \\
\\
&=& \lim _{h \to 0} \frac{\ln \left( 1+\frac{h}{x} \right)}{h} \\
\\
&=& \lim _{h \to 0} \frac{1}{x} \cdot \frac{\ln \left( 1 + \frac{h}{x} \right)}{\frac{h}{x}} \\
\\
&=& \frac{1}{x} \lim _{\varepsilon \to 0} \left( \ln (1 + \varepsilon)^{\frac{1}{\varepsilon}} \right) \\
\\
&=& \frac{1}{x}
\end{eqnarray*}

\frac{df(x)}{dx} &=& \lim _{h \to 0} \frac{\ln (x+h) – \ln x}{h}

&=& \lim _{h \to 0} \frac{\ln \left( 1+\frac{h}{x} \right)}{h}

&=& \lim _{h \to 0} \frac{1}{x} \cdot \frac{\ln \left( 1 + \frac{h}{x} \right)}{\frac{h}{x}}

&=& \frac{1}{x} \lim _{\varepsilon \to 0} \left( \ln (1 + \varepsilon)^{\frac{1}{\varepsilon}} \right)

&=& \frac{1}{x}

\begin{eqnarray*}
\frac{df(x)}{dx} &=& g (x) \\
\\
\int \frac{df(x)}{dx} dx &=& \int g (x) dx \\
\\
f(x) &=& \int g(x) dx
\end{eqnarray*}

\frac{df(x)}{dx} &=& g (x)

\int \frac{df(x)}{dx} dx &=& \int g (x) dx

f(x) &=& \int g(x) dx

$$
\int \{ \alpha f(x) + \beta g(x) \} dx = \alpha \int f(x) dx + \beta \int g(x) dx
$$

\int \{ \alpha f(x) + \beta g(x) \} dx = \alpha \int f(x) dx + \beta \int g(x) dx

$$
\int f(x) g'(x) dx = f(x) g(x) – \int f'(x) g(x) dx
$$

\int f(x) g'(x) dx = f(x) g(x) – \int f'(x) g(x) dx

\begin{eqnarray*}
\int \cos x dx &=& \sin x \\
\\
\int x \sin x dx &=& x(-\cos x) + \int 1 \cdot \cos x dx \\
\\
&=& -x \cos x + \sin x \\
\\
\int x \cos x dx &=& x \sin x + \int 1 \cdot \sin x dx \\
\\
&=& x \sin x + \cos x \\
\\
\int x^p dx &=& \frac{1}{p+1} x^{p+1} \\
\\
\int \frac{1}{x} dx &=& \ln x \\
\\
\int e^{ax} dx &=& \frac{1}{a} e^{ax}
\end{eqnarray*}

\int \cos x dx &=& \sin x

\int x \sin x dx &=& x(-\cos x) + \int 1 \cdot \cos x dx

&=& -x \cos x + \sin x

\int x \cos x dx &=& x \sin x + \int 1 \cdot \sin x dx

&=& x \sin x + \cos x

\int x^p dx &=& \frac{1}{p+1} x^{p+1}

\int \frac{1}{x} dx &=& \ln x

\int e^{ax} dx &=& \frac{1}{a} e^{ax}

\begin{eqnarray*}
\int_{0}^{2\pi} \sin x dx &=& \Bigl[\cos x \Bigr]_{0}^{2\pi} = 1-1 = 0 \\
\\
\int_{0}^{2\pi} \sin ^2 x dx &=& \frac{1}{2} \int_{0}^{2\pi} (1 – \cos 2x) dx = \frac{1}{2} \Bigl[x \Bigr] _{0}^{2\pi} =\pi
\end{eqnarray*}

\int_{0}^{2\pi} \sin x dx &=& \Bigl[\cos x \Bigr]_{0}^{2\pi} = 1-1 = 0

\int_{0}^{2\pi} \sin ^2 x dx &=& \frac{1}{2} \int_{0}^{2\pi} (1 – \cos 2x) dx = \frac{1}{2} \Bigl[x \Bigr] _{0}^{2\pi} =\pi

\begin{eqnarray*}
f(x) &=& f(a) + f'(a) (x-a) + \frac{1}{2!} f"(a) (x-a)^2 + \cdots \\
\\
&=& \sum _{n=0}^{\infty} \frac{1}{n!} f^{(n)} (a)(x-a)^n
\end{eqnarray*}

f(x) &=& f(a) + f'(a) (x-a) + \frac{1}{2!} f"(a) (x-a)^2 + \cdots

&=& \sum _{n=0}^{\infty} \frac{1}{n!} f^{(n)} (a)(x-a)^n

\begin{eqnarray*}
(1+x)^n &=& 1+nx +\frac{1}{2} n(n-1) x^2 + \cdots
= \sum _{n=0}^{\infty} \frac{n(n-1) \cdots (n-k+1)}{k!} x^k \ \ (|x| < 1) \\ \\ e^x &=& 1+x+\frac{1}{2!} x^2 + \cdots = \sum _{n=0}^{\infty} \frac{1}{n!} x^n\\ \\ \sin x &=& x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \cdots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} \\ \\ \cos x &=& 1 - \frac{1}{2!} x^2 + \frac{1}{4!} x^4 - \cdots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} \\ \\ \ln (1+x) &=& x - \frac{1}{2} x^2 + \frac{1}{3} x^3 - \cdots = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n} x^n \ \ (-1 < x \le 1) \\ \end{eqnarray*}

(1+x)^n &=& 1+nx +\frac{1}{2} n(n-1) x^2 + \cdots
= \sum _{n=0}^{\infty} \frac{n(n-1) \cdots (n-k+1)}{k!} x^k \ \ (|x| < 1) e^x &=& 1+x+\frac{1}{2!} x^2 + \cdots = \sum _{n=0}^{\infty} \frac{1}{n!} x^n \sin x &=& x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \cdots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} \cos x &=& 1 - \frac{1}{2!} x^2 + \frac{1}{4!} x^4 - \cdots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} \ln (1+x) &=& x - \frac{1}{2} x^2 + \frac{1}{3} x^3 - \cdots = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n} x^n \ \ (-1 < x \le 1)

\begin{eqnarray*}
\frac{\partial f}{\partial x} &=& \lim _{h \to 0} \frac{f(x+h,y) – f(x,y)}{h} = \left( \frac{\partial f}{\partial x} \right)_y \\
\\
\frac{\partial f}{\partial y} &=& \lim _{h \to 0} \frac{f(x,y+h) – f(x,y)}{h} = \left( \frac{\partial f}{\partial y} \right)_x\\
\end{eqnarray*}

\frac{\partial f}{\partial x} &=& \lim _{h \to 0} \frac{f(x+h,y) – f(x,y)}{h} = \left( \frac{\partial f}{\partial x} \right)_y

\frac{\partial f}{\partial y} &=& \lim _{h \to 0} \frac{f(x,y+h) – f(x,y)}{h} = \left( \frac{\partial f}{\partial y} \right)_x

\begin{eqnarray*}
\Delta f &=& f(x + \Delta x, y + \Delta y) – f(x,y) \\
\\
f(x + \Delta x, y + \Delta y) &=& f(x,y + \Delta y) + \frac{\partial f}{\partial x} \Delta x + \cdots \\
\\
&=& f(x,y) + \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y + \cdots \\
\\
\Delta f &=& \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y
\end{eqnarray*}

\Delta f &=& f(x + \Delta x, y + \Delta y) – f(x,y)

f(x + \Delta x, y + \Delta y) &=& f(x,y + \Delta y) + \frac{\partial f}{\partial x} \Delta x + \cdots

&=& f(x,y) + \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y + \cdots

\Delta f &=& \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y