付録

Table of Derivatives微分公式表

General Formulas一般的な公式

\frac{d}{d x} (c) = 0

\frac{d}{d x} (c) = 0

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

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

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

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

\frac{d}{d x} (x^{n}) = n x^{n - 1}, for real numbers n

\frac{d}{d x} (x^{n}) = n x^{n - 1}, 実数 n

\frac{d}{d x} (c f (x)) = c f^{'} (x)

\frac{d}{d x} (c f (x)) = c f^{'} (x)

\frac{d}{d x} (f (x) - g (x)) = f^{'} (x) - g^{'} (x)

\frac{d}{d x} (f (x) - g (x)) = f^{'} (x) - g^{'} (x)

\frac{d}{d x} (\frac{f (x)}{g (x)}) = \frac{g (x) f^{'} (x) - f (x) g^{'} (x)}{{(g (x))}^{2}}

\frac{d}{d x} (\frac{f (x)}{g (x)}) = \frac{g (x) f^{'} (x) - f (x) g^{'} (x)}{{(g (x))}^{2}}

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

Trigonometric Functions三角関数

\frac{d}{d x} (sin x) = cos x

\frac{d}{d x} (sin x) = cos x

10.

\frac{d}{d x} (tan x) = {sec}^{2} x

10.

\frac{d}{d x} (tan x) = {sec}^{2} x

11.

\frac{d}{d x} (sec x) = sec x tan x

11.

\frac{d}{d x} (sec x) = sec x tan x

12.

\frac{d}{d x} (cos x) = − sin x

12.

\frac{d}{d x} (cos x) = − sin x

13.

\frac{d}{d x} (cot x) = − {csc}^{2} x

13.

\frac{d}{d x} (cot x) = − {csc}^{2} x

14.

\frac{d}{d x} (csc x) = −csc x cot x

14.

\frac{d}{d x} (csc x) = −csc x cot x

Inverse Trigonometric Functions逆三角関数

15.

\frac{d}{d x} ({sin}^{−1} x) = \frac{1}{\sqrt{1 - x^{2}}}

15.

\frac{d}{d x} ({sin}^{−1} x) = \frac{1}{\sqrt{1 - x^{2}}}

16.

\frac{d}{d x} ({tan}^{−1} x) = \frac{1}{1 + x^{2}}

16.

\frac{d}{d x} ({tan}^{−1} x) = \frac{1}{1 + x^{2}}

17.

\frac{d}{d x} ({sec}^{−1} x) = \frac{1}{| x | \sqrt{x^{2} - 1}}

17.

\frac{d}{d x} ({sec}^{−1} x) = \frac{1}{| x | \sqrt{x^{2} - 1}}

18.

\frac{d}{d x} ({cos}^{−1} x) = - \frac{1}{\sqrt{1 - x^{2}}}

18.

\frac{d}{d x} ({cos}^{−1} x) = - \frac{1}{\sqrt{1 - x^{2}}}

19.

\frac{d}{d x} ({cot}^{−1} x) = - \frac{1}{1 + x^{2}}

19.

\frac{d}{d x} ({cot}^{−1} x) = - \frac{1}{1 + x^{2}}

20.

\frac{d}{d x} ({csc}^{−1} x) = - \frac{1}{| x | \sqrt{x^{2} - 1}}

20.

\frac{d}{d x} ({csc}^{−1} x) = - \frac{1}{| x | \sqrt{x^{2} - 1}}

Exponential and Logarithmic Functions指数関数・対数関数

21.

\frac{d}{d x} (e^{x}) = e^{x}

21.

\frac{d}{d x} (e^{x}) = e^{x}

22.

\frac{d}{d x} (ln | x |) = \frac{1}{x}

22.

\frac{d}{d x} (ln | x |) = \frac{1}{x}

23.

\frac{d}{d x} (b^{x}) = b^{x} ln b

23.

\frac{d}{d x} (b^{x}) = b^{x} ln b

24.

\frac{d}{d x} ({log}_{b} x) = \frac{1}{x ln b}

24.

\frac{d}{d x} ({log}_{b} x) = \frac{1}{x ln b}

Hyperbolic Functions双曲線関数

25.

\frac{d}{d x} (sinh x) = cosh x

25.

\frac{d}{d x} (sinh x) = cosh x

26.

\frac{d}{d x} (tanh x) = {sech}^{2} x

26.

\frac{d}{d x} (tanh x) = {sech}^{2} x

27.

\frac{d}{d x} (sech x) = −sech x tanh x

27.

\frac{d}{d x} (sech x) = −sech x tanh x

28.

\frac{d}{d x} (cosh x) = sinh x

28.

\frac{d}{d x} (cosh x) = sinh x

29.

\frac{d}{d x} (coth x) = − {csch}^{2} x

29.

\frac{d}{d x} (coth x) = − {csch}^{2} x

30.

\frac{d}{d x} (csch x) = −csch x coth x

30.

\frac{d}{d x} (csch x) = −csch x coth x

Inverse Hyperbolic Functions逆双曲線関数

31.

\frac{d}{d x} ({sinh}^{−1} x) = \frac{1}{\sqrt{x^{2} + 1}}

31.

\frac{d}{d x} ({sinh}^{−1} x) = \frac{1}{\sqrt{x^{2} + 1}}

32.

\frac{d}{d x} ({tanh}^{−1} x) = \frac{1}{1 - x^{2}} (| x | < 1)

32.

\frac{d}{d x} ({tanh}^{−1} x) = \frac{1}{1 - x^{2}} (| x | < 1)

33.

\frac{d}{d x} ({sech}^{−1} x) = - \frac{1}{x \sqrt{1 - x^{2}}} (0 < x < 1)

33.

\frac{d}{d x} ({sech}^{−1} x) = - \frac{1}{x \sqrt{1 - x^{2}}} (0 < x < 1)

34.

\frac{d}{d x} ({cosh}^{−1} x) = \frac{1}{\sqrt{x^{2} - 1}} (x > 1)

34.

\frac{d}{d x} ({cosh}^{−1} x) = \frac{1}{\sqrt{x^{2} - 1}} (x > 1)

35.

\frac{d}{d x} ({coth}^{−1} x) = \frac{1}{1 - x^{2}} (| x | > 1)

35.

\frac{d}{d x} ({coth}^{−1} x) = \frac{1}{1 - x^{2}} (| x | > 1)

36.

\frac{d}{d x} ({csch}^{−1} x) = - \frac{1}{| x | \sqrt{1 + x^{2}}} (x \neq 0)

36.

\frac{d}{d x} ({csch}^{−1} x) = - \frac{1}{| x | \sqrt{1 + x^{2}}} (x \neq 0)