导数列表

以下的列表列出了许多函数的导数。f 和g是可微函数，而别的皆为常数。用这些公式，可以求出任何初等函数的导数。 線性法则 d ( M f ) d x = M d f d x ; [ M f ( x ) ] ′ = M f ′ ( x ) {\displaystyle {{\mbox{d}}(Mf) \over {\mbox{d}}x}=M{{\mbox{d}}f \over {\mbox{d}}x};\qquad [Mf(x)]'=Mf'(x)} d ( f ± g ) d x = d f d x ± d g d x   {\displaystyle {{{\mbox{d}}(f\pm g)} \over {{\mbox{d}}x}}={{\mbox{d}}f \over {\mbox{d}}x}\pm {{\mbox{d}}g \over {\mbox{d}}x}\ } 乘法定则 d f g d x = d f d x g + f d g d x {\displaystyle {{\mbox{d}}fg \over {\mbox{d}}x}={{\mbox{d}}f \over {\mbox{d}}x}g+f{\frac {{\mbox{d}}g}{{\mbox{d}}x}}} 除法定则 d f g d x = d f d x g − f d g d x g 2 ( g ≠ 0 ) {\displaystyle {\frac {{\mbox{d}}{\dfrac {f}{g}}}{{\mbox{d}}x}}={\frac {{\dfrac {{\mbox{d}}f}{{\mbox{d}}x}}g-f{\dfrac {{\mbox{d}}g}{{\mbox{d}}x}}}{g^{2}}}\qquad (g\neq 0)} 倒数定则 d 1 g d x = − d g d x g 2 ( g ≠ 0 ) {\displaystyle {\frac {{\mbox{d}}{\dfrac {1}{g}}}{{\mbox{d}}x}}={\frac {-{\dfrac {{\mbox{d}}g}{{\mbox{d}}x}}}{g^{2}}}\qquad (g\neq 0)} 复合函数求导法则（連鎖定則） ( f ∘ g ) ′ ( x ) = f ′ ( g ( x ) ) g ′ ( x ) . {\displaystyle (f\circ g)'(x)=f'(g(x))g'(x).} d f [ g ( x ) ] d x = d f ( g ) d g d g d x = f ′ [ g ( x ) ] g ′ ( x ) {\displaystyle {\frac {{\mbox{d}}f[g(x)]}{{\mbox{d}}x}}={\frac {{\mbox{d}}f(g)}{{\mbox{d}}g}}{\frac {{\mbox{d}}g}{{\mbox{d}}x}}=f'[g(x)]g'(x)} 反函数的导数 由于 g ( f ( x ) ) = x {\displaystyle g(f(x))=x} ，故 g ( f ( x ) ) ′ = 1 {\displaystyle g(f(x))'=1} ，根據复合函数求导法则，則 g ( f ( x ) ) ′ = d g [ f ( x ) ] d x = d g ( f ) d f d f d x = 1 {\displaystyle g(f(x))'={\frac {{\mbox{d}}g[f(x)]}{{\mbox{d}}x}}={\frac {{\mbox{d}}g(f)}{{\mbox{d}}f}}{\frac {{\mbox{d}}f}{{\mbox{d}}x}}=1} 所以 d f d x = 1 d g ( f ) d f = [ d g ( f ) d f ] − 1 = [ g ′ ( f ) ] − 1 {\displaystyle {\frac {{\mbox{d}}f}{{\mbox{d}}x}}={\frac {1}{\dfrac {{\mbox{d}}g(f)}{{\mbox{d}}f}}}=[{\frac {{\mbox{d}}g(f)}{{\mbox{d}}f}}]^{-1}=[g'(f)]^{-1}} 同理 d g d x = 1 d f ( g ) d g = [ d f ( g ) d g ] − 1 = [ f ′ ( g ) ] − 1 {\displaystyle {\frac {{\mbox{d}}g}{{\mbox{d}}x}}={\frac {1}{\dfrac {{\mbox{d}}f(g)}{{\mbox{d}}g}}}=[{\frac {{\mbox{d}}f(g)}{{\mbox{d}}g}}]^{-1}=[f'(g)]^{-1}} 广义幂法则 ( f g ) ′ = ( e g ln ⁡ f ) ′ = f g ( g ′ ln ⁡ f + g f f ′ ) {\displaystyle (f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(g'\ln f+{\frac {g}{f}}f'\right)} (n为任意实常数) d n d x = 0 {\displaystyle {{\mbox{d}}n \over {\mbox{d}}x}=0} d x d x = 1 {\displaystyle {{\mbox{d}}x \over {\mbox{d}}x}=1} d x n d x = n x n − 1 {\displaystyle {{\mbox{d}}x^{n} \over {\mbox{d}}x}=nx^{n-1}\qquad } 當 n ≤ 1 {\displaystyle n\leq 1} ，則 x ≠ 0 {\displaystyle x\neq 0} d | x | d x = x | x | = | x | x = sgn ⁡ x x ≠ 0 {\displaystyle {{\mbox{d}}|x| \over {\mbox{d}}x}={x \over |x|}={|x| \over x}=\operatorname {sgn} x\qquad x\neq 0} d d x e f ( x ) = f ′ ( x ) e f ( x ) {\displaystyle {d \over dx}e^{f(x)}=f'(x)e^{f(x)}} . d   e x d x = lim Δ x → 0 e x − e x − Δ x Δ x = e x lim Δ x → 0 1 − e − Δ x Δ x = e x {\displaystyle {\begin{aligned}{\frac {{\mbox{d}}\ e^{x}}{{\mbox{d}}x}}&=\lim _{\Delta x\to 0}{\frac {e^{x}-e^{x-\Delta x}}{\Delta x}}\\&=e^{x}\lim _{\Delta x\to 0}{\frac {1-e^{-\Delta x}}{\Delta x}}\\&=e^{x}\end{aligned}}} d   α x d x = d   e x ln α d x = d e x ln α d   x ln α ⋅ d   x ln α d x = e x ln α ln α = α x ln α {\displaystyle {\begin{aligned}{\frac {{\mbox{d}}\ \alpha ^{x}}{{\mbox{d}}x}}&={\frac {{\mbox{d}}\ e^{x\!\ln \!\alpha }}{{\mbox{d}}x}}\\&={\frac {{\mbox{d}}e^{x\!\ln \!\alpha }}{{\mbox{d}}\ x\!\ln \!\alpha }}\cdot {\frac {{\mbox{d}}\ x\!\ln \!\alpha }{{\mbox{d}}x}}\\&=e^{x\!\ln \!\alpha }\!\ln \!\alpha \\&=\alpha ^{x}\!\ln \!\alpha \end{aligned}}} d ln ⁡ x d x = lim h → 0 ln ⁡ ( x + h ) − ln ⁡ x h = lim h → 0 ( 1 h ln ⁡ ( x + h x ) ) = lim h → 0 ( x x h ln ⁡ ( 1 + h x ) ) = 1 x ln ⁡ ( lim h → 0 ( 1 + h x ) x h ) = 1 x ln ⁡ e = 1 x {\displaystyle {\begin{aligned}{\frac {{\mbox{d}}\ln x}{{\mbox{d}}x}}&=\lim _{h\to 0}{\frac {\ln(x+h)-\ln x}{h}}\\&=\lim _{h\to 0}({\frac {1}{h}}\ln({\frac {x+h}{x}}))\\&=\lim _{h\to 0}({\frac {x}{xh}}\ln(1+{\frac {h}{x}}))\\&={\frac {1}{x}}\ln(\lim _{h\to 0}(1+{\frac {h}{x}})^{\frac {x}{h}})\\&={\frac {1}{x}}\ln e\\&={\frac {1}{x}}\end{aligned}}} d log α ⁡ | x | d x = 1 ln ⁡ α d ln ⁡ | x | d x = 1 x ln ⁡ α {\displaystyle {\frac {{\mbox{d}}\log _{\alpha }|x|}{{\mbox{d}}x}}={1 \over \ln \alpha }{\frac {{\mbox{d}}\ln |x|}{{\mbox{d}}x}}={1 \over x\ln \alpha }} d   x x d x = x x ( 1 + ln ⁡ x ) {\displaystyle {\frac {{\mbox{d}}\ x^{x}}{{\mbox{d}}x}}=x^{x}(1+\ln x)} ( sin ⁡ x ) ′ = lim h → 0 sin ⁡ ( x + h ) − sin ⁡ x h = lim h → 0 sin ⁡ x cos ⁡ h + cos ⁡ x sin ⁡ h − sin ⁡ x h = lim h → 0 ( sin ⁡ x cos ⁡ h − 1 h + cos ⁡ x sin ⁡ h h ) = cos ⁡ x {\displaystyle {\begin{aligned}(\sin 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}(\sin x{\frac {\cos h-1}{h}}+\cos x{\frac {\sin h}{h}})\\&=\cos x\end{aligned}}} ( cos ⁡ x ) ′ = lim h → 0 cos ⁡ ( x + h ) − cos ⁡ x h = lim h → 0 cos ⁡ x cos ⁡ h − sin ⁡ x sin ⁡ h − cos ⁡ x h = lim h → 0 ( cos ⁡ x cos ⁡ h − 1 h − sin ⁡ x sin ⁡ h h ) = − sin ⁡ x {\displaystyle {\begin{aligned}(\cos 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}(\cos x{\frac {\cos h-1}{h}}-\sin x{\frac {\sin h}{h}})\\&=-\sin x\end{aligned}}} ( tan ⁡ x ) ′ = ( sin ⁡ x cos ⁡ x ) ′ = ( sin ⁡ x ) ′ cos ⁡ x − sin ⁡ x ( cos ⁡ x ) ′ cos 2 ⁡ x = cos 2 ⁡ x + sin 2 ⁡ x cos 2 ⁡ x = 1 cos 2 ⁡ x = sec 2 ⁡ x {\displaystyle {\begin{aligned}(\tan x)'&=({\frac {\sin x}{\cos x}})'\\&={\frac {(\sin x)'\cos x-\sin x(\cos x)'}{\cos ^{2}x}}\\&={\frac {\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}}\\&={\frac {1}{\cos ^{2}x}}=\sec ^{2}x\end{aligned}}} ( cot ⁡ x ) ′ = ( cos ⁡ x sin ⁡ x ) ′ = ( cos ⁡ x ) ′ sin ⁡ x − cos ⁡ x ( sin ⁡ x ) ′ sin 2 ⁡ x = − sin 2 ⁡ x − cos 2 ⁡ x sin 2 ⁡ x = − 1 sin 2 ⁡ x = − csc 2 ⁡ x {\displaystyle {\begin{aligned}(\cot x)'&=({\frac {\cos x}{\sin x}})'\\&={\frac {(\cos x)'\sin x-\cos x(\sin x)'}{\sin ^{2}x}}\\&={\frac {-\sin ^{2}x-\cos ^{2}x}{\sin ^{2}x}}\\&=-{\frac {1}{\sin ^{2}x}}=-\csc ^{2}x\end{aligned}}} ( sec ⁡ x ) ′ = ( 1 cos ⁡ x ) ′ = sin ⁡ x cos 2 ⁡ x = sec ⁡ x tan ⁡ x {\displaystyle {\begin{aligned}(\sec x)'&=({\frac {1}{\cos x}})'\\&={\frac {\sin x}{\cos ^{2}x}}\\&=\sec x\tan x\end{aligned}}} ( csc ⁡ x ) ′ = ( 1 sin ⁡ x ) ′ = − cos ⁡ x sin 2 ⁡ x = − csc ⁡ x cot ⁡ x {\displaystyle {\begin{aligned}(\csc x)'&=({\frac {1}{\sin x}})'\\&={\frac {-\cos x}{\sin ^{2}x}}\\&=-\csc x\cot x\end{aligned}}} ( arcsin ⁡ x ) ′ = 1 cos ⁡ ( arcsin ⁡ x ) ⇔ sin ⁡ ( arcsin ⁡ x ) = x ⇔ cos ⁡ ( arcsin ⁡ x ) ( arcsin ⁡ x ) ′ = 1 = 1 1 − sin 2 ⁡ ( arcsin ⁡ x ) = 1 1 − x 2     ( | x | < 1 ) {\displaystyle {\begin{aligned}(\arcsin x)'&={\frac {1}{\cos(\arcsin x)}}\Leftrightarrow \sin(\arcsin x)=x\Leftrightarrow \cos(\arcsin x)(\arcsin x)'=1\\&={\frac {1}{\sqrt {1-\sin ^{2}(\arcsin x)}}}\\&={\frac {1}{\sqrt {1-x^{2}}}}\ \ (\left|x\right|<1)\end{aligned}}} ( arccos ⁡ x ) ′ = 1 − sin ⁡ ( arccos ⁡ x ) ⇔ cos ⁡ ( arccos ⁡ x ) = x ⇔ − sin ⁡ ( arccos ⁡ x ) ( arccos ⁡ x ) ′ = 1 = − 1 1 − cos 2 ⁡ ( arccos ⁡ x ) = − 1 1 − x 2     ( | x | < 1 ) {\displaystyle {\begin{aligned}(\arccos x)'&={\frac {1}{-\sin(\arccos x)}}\Leftrightarrow \cos(\arccos x)=x\Leftrightarrow -\sin(\arccos x)(\arccos x)'=1\\&=-{\frac {1}{\sqrt {1-\cos ^{2}(\arccos x)}}}\\&=-{\frac {1}{\sqrt {1-x^{2}}}}\ \ (\left|x\right|<1)\end{aligned}}} ( arctan ⁡ x ) ′ = 1 sec 2 ⁡ ( arctan ⁡ x ) ⇔ tan ⁡ ( arctan ⁡ x ) = x ⇔ sec 2 ⁡ ( arctan ⁡ x ) ( arctan ⁡ x ) ′ = 1 = 1 1 + tan 2 ⁡ ( arctan ⁡ x ) = 1 1 + x 2 {\displaystyle {\begin{aligned}(\arctan x)'&={\frac {1}{\sec ^{2}(\arctan x)}}\Leftrightarrow \tan(\arctan x)=x\Leftrightarrow \sec ^{2}(\arctan x)(\arctan x)'=1\\&={\frac {1}{1+\tan ^{2}(\arctan x)}}\\&={\frac {1}{1+x^{2}}}\end{aligned}}} ( arccot ⁡ x ) ′ = 1 − csc 2 ⁡ ( arccot ⁡ x ) ⇔ cot ⁡ ( arccot ⁡ x ) = x ⇔ − csc 2 ⁡ ( arccot ⁡ x ) ( arccot ⁡ x ) ′ = 1 = − 1 1 + cot 2 ⁡ ( arccot ⁡ x ) = − 1 1 + x 2 {\displaystyle {\begin{aligned}(\operatorname {arccot} x)'&={\frac {1}{-\csc ^{2}(\operatorname {arccot} x)}}\Leftrightarrow \cot(\operatorname {arccot} x)=x\Leftrightarrow -\csc ^{2}(\operatorname {arccot} x)(\operatorname {arccot} x)'=1\\&=-{\frac {1}{1+\cot ^{2}(\operatorname {arccot} x)}}\\&=-{\frac {1}{1+x^{2}}}\end{aligned}}} ( arcsec ⁡ x ) ′ = 1 sec ⁡ ( arcsec ⁡ x ) tan ⁡ ( arcsec ⁡ x ) ⇔ sec ⁡ ( arcsec ⁡ x ) = x ⇔ sec ⁡ ( arcsec ⁡ x ) tan ⁡ ( arcsec ⁡ x ) ( arcsec ⁡ x ) ′ = 1 = 1 | x | sec 2 ⁡ ( arcsec ⁡ x ) − 1 = 1 | x | x 2 − 1     ( | x | > 1 ) {\displaystyle {\begin{aligned}(\operatorname {arcsec} x)'&={\frac {1}{\sec(\operatorname {arcsec} x)\tan(\operatorname {arcsec} x)}}\Leftrightarrow \sec(\operatorname {arcsec} x)=x\Leftrightarrow \sec(\operatorname {arcsec} x)\tan(\operatorname {arcsec} x)(\operatorname {arcsec} x)'=1\\&={\frac {1}{|x|{\sqrt {\sec ^{2}(\operatorname {arcsec} x)-1}}}}\\&={\frac {1}{|x|{\sqrt {x^{2}-1}}}}\ \ (\left|x\right|>1)\end{aligned}}} ( arccsc ⁡ x ) ′ = 1 − csc ⁡ ( arccsc ⁡ x ) cot ⁡ ( arccsc ⁡ x ) ⇔ csc ⁡ ( arccsc ⁡ x ) = x ⇔ − csc ⁡ ( arccsc ⁡ x ) cot ⁡ ( arccsc ⁡ x ) ( arccsc ⁡ x ) ′ = 1 = − 1 | x | csc 2 ⁡ ( arcsec ⁡ x ) − 1 = − 1 | x | x 2 − 1     ( | x | > 1 ) {\displaystyle {\begin{aligned}(\operatorname {arccsc} x)'&={\frac {1}{-\csc(\operatorname {arccsc} x)\cot(\operatorname {arccsc} x)}}\Leftrightarrow \csc(\operatorname {arccsc} x)=x\Leftrightarrow -\csc(\operatorname {arccsc} x)\cot(\operatorname {arccsc} x)(\operatorname {arccsc} x)'=1\\&=-{\frac {1}{|x|{\sqrt {\csc ^{2}(\operatorname {arcsec} x)-1}}}}\\&=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}\ \ (\left|x\right|>1)\end{aligned}}} 這是前述廣義冪法則在「 f ( x ) = α {\displaystyle f(x)=\alpha } 且 g ( x ) = x {\displaystyle g(x)=x} 」時的特例。 這是前述廣義冪法則在「 f ( x ) = x {\displaystyle f(x)=x} 且 g ( x ) = x {\displaystyle g(x)=x} 」時的特例。