d d x [ f ( x ) ± g ( x ) ] = d d x f ( x ) ± d d x g ( x ) {\displaystyle {\frac {d}{dx}}[f(x)\pm g(x)]={\frac {d}{dx}}f(x)\pm {\frac {d}{dx}}g(x)}
d d x [ c ⋅ f ( x ) ] = c ⋅ [ d d x f ( x ) ] {\displaystyle {\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}}
d d x [ f ( x ) ⋅ g ( x ) ] = f ( x ) ⋅ [ d d x g ( x ) ] + g ( x ) ⋅ [ d d x f ( x ) ] {\displaystyle {\frac {d}{dx}}[f(x)\cdot g(x)]=f(x)\cdot {\bigg [}{\frac {d}{dx}}g(x){\bigg ]}+g(x)\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}}
d d x [ f ( x ) g ( x ) ] = g ( x ) ⋅ [ d d x f ( x ) ] − f ( x ) ⋅ [ d d x g ( x ) ] g ( x ) 2 {\displaystyle {\frac {d}{dx}}\left[{\frac {f(x)}{g(x)}}\right]={\frac {g(x)\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}-f(x)\cdot {\bigg [}{\frac {d}{dx}}g(x){\bigg ]}}{g(x)^{2}}}}
d d x [ f ( g ( x ) ) ] = f ′ ( g ( x ) ) ⋅ g ′ ( x ) {\displaystyle {\frac {d}{dx}}\left[f(g(x))\right]=f'(g(x))\cdot g'(x)}
d d x ( c ) = 0 {\displaystyle {\frac {d}{dx}}(c)=0}
d d x x = 1 {\displaystyle {\frac {d}{dx}}x=1}
d d x x n = n x n − 1 {\displaystyle {\frac {d}{dx}}x^{n}=nx^{n-1}}
d d x x = 1 2 x {\displaystyle {\frac {d}{dx}}{\sqrt {x}}={\frac {1}{2{\sqrt {x}}}}}
d d x 1 x = − 1 x 2 {\displaystyle {\frac {d}{dx}}{\frac {1}{x}}=-{\frac {1}{x^{2}}}}
d d x sin ( x ) = cos ( x ) {\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)}
d d x cos ( x ) = − sin ( x ) {\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x)}
d d x tan ( x ) = sec 2 ( x ) {\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)}
d d x cot ( x ) = − csc 2 ( x ) {\displaystyle {\frac {d}{dx}}\cot(x)=-\csc ^{2}(x)}
d d x sec ( x ) = sec ( x ) ⋅ tan ( x ) {\displaystyle {\frac {d}{dx}}\sec(x)=\sec(x)\cdot \tan(x)}
d d x csc ( x ) = − csc ( x ) ⋅ cot ( x ) {\displaystyle {\frac {d}{dx}}\csc(x)=-\csc(x)\cdot \cot(x)}
d d x arcsin ( x ) = 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}}
d d x arccos ( x ) = − 1 1 − x 2 {\displaystyle {\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}}}
d d x arctan ( x ) = 1 1 + x 2 {\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}}
d d x arcsec ( x ) = 1 | x | x 2 − 1 {\displaystyle {\frac {d}{dx}}\operatorname {arcsec}(x)={\frac {1}{|x|{\sqrt {x^{2}-1}}}}}
d d x arccot ( x ) = − 1 1 + x 2 {\displaystyle {\frac {d}{dx}}\operatorname {arccot}(x)=-{\frac {1}{1+x^{2}}}}
d d x arccsc ( x ) = − 1 | x | x 2 − 1 {\displaystyle {\frac {d}{dx}}\operatorname {arccsc}(x)=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}}
d d x sinh ( x ) = cosh ( x ) {\displaystyle {\frac {d}{dx}}\sinh(x)=\cosh(x)}
d d x cosh ( x ) = sinh ( x ) {\displaystyle {\frac {d}{dx}}\cosh(x)=\sinh(x)}
d d x tanh ( x ) = sech 2 ( x ) {\displaystyle {\frac {d}{dx}}\tanh(x)={\mbox{sech}}^{2}(x)}
d d x sech ( x ) = − tanh ( x ) ⋅ sech ( x ) {\displaystyle {\frac {d}{dx}}{\mbox{sech}}(x)=-\tanh(x)\cdot {\mbox{sech}}(x)}
d d x coth ( x ) = − csch 2 ( x ) {\displaystyle {\frac {d}{dx}}{\mbox{coth}}(x)=-{\mbox{csch}}^{2}(x)}
d d x csch ( x ) = − coth ( x ) ⋅ csch ( x ) {\displaystyle {\frac {d}{dx}}{\mbox{csch}}(x)=-{\mbox{coth}}(x)\cdot {\mbox{csch}}(x)}
d d x e x = e x {\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}
d d x a x = a x ⋅ ln ( a ) if a > 0 {\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\cdot \ln(a)\qquad {\mbox{if }}a>0}
d d x ln ( | x | ) = 1 x {\displaystyle {\frac {d}{dx}}\ln(|x|)={\frac {1}{x}}}
d d x log a ( x ) = 1 x ln ( a ) if a > 0 , a ≠ 1 {\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}}\qquad {\mbox{if }}a>0,a\neq 1}
d d x [ f ( x ) g ( x ) ] = d d x [ e g ( x ) ⋅ ln [ f ( x ) ] ] = f ( x ) g ( x ) ⋅ ( g ( x ) f ( x ) ⋅ [ d d x f ( x ) ] + ln [ f ( x ) ] ⋅ [ d d x g ( x ) ] ) , f > 0 {\displaystyle {\frac {d}{dx}}[f(x)^{g(x)}]={\frac {d}{dx}}\left[e^{g(x)\cdot \ln[f(x)]}\right]=f(x)^{g(x)}\cdot \left({\frac {g(x)}{f(x)}}\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}+\ln[f(x)]\cdot {\bigg [}{\frac {d}{dx}}g(x){\bigg ]}\right),\qquad f>0}
d d x c f ( x ) = d d x ( e f ( x ) ⋅ ln ( c ) ) = c f ( x ) ⋅ ln ( c ) ⋅ [ d d x f ( x ) ] {\displaystyle {\frac {d}{dx}}c^{f(x)}={\frac {d}{dx}}\left(e^{f(x)\cdot \ln(c)}\right)=c^{f(x)}\cdot \ln(c)\cdot \left[{\frac {d}{dx}}f(x)\right]}
![{\displaystyle {\frac {d}{dx}}[f(x)^{g(x)}]={\frac {d}{dx}}\left[e^{g(x)\cdot \ln[f(x)]}\right]=f(x)^{g(x)}\cdot \left({\frac {g(x)}{f(x)}}\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}+\ln[f(x)]\cdot {\bigg [}{\frac {d}{dx}}g(x){\bigg ]}\right),\qquad f>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c12f1cb9d233afe53dc10090197121351d634dbb)
![{\displaystyle {\frac {d}{dx}}c^{f(x)}={\frac {d}{dx}}\left(e^{f(x)\cdot \ln(c)}\right)=c^{f(x)}\cdot \ln(c)\cdot \left[{\frac {d}{dx}}f(x)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/461ca447f11ae9fba2d34aa079cd357c8e0ea930)
![{\displaystyle {\frac {d}{dx}}[f(x)\pm g(x)]={\frac {d}{dx}}f(x)\pm {\frac {d}{dx}}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74e449d337a2b4f202ac0ca5e06a166e3af6a758)
![{\displaystyle {\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aadafadb427168f231f5aae42402d50e05654b6a)
![{\displaystyle {\frac {d}{dx}}[f(x)\cdot g(x)]=f(x)\cdot {\bigg [}{\frac {d}{dx}}g(x){\bigg ]}+g(x)\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77689f2c9431250f445ed3e580d1a25e03f014de)
![{\displaystyle {\frac {d}{dx}}\left[{\frac {f(x)}{g(x)}}\right]={\frac {g(x)\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}-f(x)\cdot {\bigg [}{\frac {d}{dx}}g(x){\bigg ]}}{g(x)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/feaef9dfded3a1256bc380947b5838e9cf7fdcf7)
![{\displaystyle {\frac {d}{dx}}\left[f(g(x))\right]=f'(g(x))\cdot g'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c352eb1e31ced1dbe5ff6cd1d0531acbb4d6895)
