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[[uk:Таблиця похідних]] |
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[[uk:Таблиця похідних]] |
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ALBERTO |
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O ALBERTO É O MAIOR! |
Revisão das 15h50min de 9 de janeiro de 2009
A operação primária do cálculo diferencial é encontrar a derivada de uma função. Na tabela a seguir, f e g são diferenciáveis em
, e c é um número real. Essas fórmulas são suficientes pap diferenciar qualquer função elementar.
Regras gerais de diferenciação
- Linearidade
![{\displaystyle \left({cf}\right)'=cf'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39da2451f330fae32b6ba571000d159d36cb0c46)
![{\displaystyle \left({f+g}\right)'=f'+g'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a356fcc9b7f291689d9de9086dd2fc12ed1af479)
- Regra do produto
![{\displaystyle \left({fg}\right)'=f'g+fg'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b087f14a89a896de41077c78ad39da0a56412e97)
- Regra do quociente
![{\displaystyle \left({f \over g}\right)'={f'g-fg' \over g^{2}},\qquad g\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a31aa870daebebcde7a311f0e0e536ec749fa04)
- Regra da cadeia
![{\displaystyle (f\circ g)'=(f'\circ g)g'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8f37ab4032a1466a59d9f3c70e2e669bc5b560)
Derivadas de funções simples
![{\displaystyle {d \over dx}c=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d66384c0fe25d80d440b317f08c9b8be9253e77)
![{\displaystyle {d \over dx}x=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03e3d9ed50d0216e5b16c4827596e3fdcc2deacb)
![{\displaystyle {d \over dx}cx=c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/867347efd790e06d7298f9a61aea5041e41fb86d)
![{\displaystyle {d \over dx}|x|={|x| \over x}=\operatorname {sgn} x,\qquad x\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350ec1c5235e07d0aaa3ffadbf597273a1caf291)
![{\displaystyle {d \over dx}x^{c}=cx^{c-1}\qquad {\mbox{ onde tanto }}x^{c}{\mbox{ quanto }}cx^{c-1}{\mbox{ sao definidos}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/342bf5ff0652c7ea1b16dcd681dbea010177d09c)
![{\displaystyle {d \over dx}\left({1 \over x}\right)={d \over dx}\left(x^{-1}\right)=-x^{-2}=-{1 \over x^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e841696967b58af8cc0eb528e604a7df0e52cdbd)
![{\displaystyle {d \over dx}\left({1 \over x^{c}}\right)={d \over dx}\left(x^{-c}\right)=-{c \over x^{c+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5608969750f40ea606c7dea0f257c76014f81895)
![{\displaystyle {d \over dx}{\sqrt {x}}={d \over dx}x^{1 \over 2}={1 \over 2}x^{-{1 \over 2}}={1 \over 2{\sqrt {x}}},\qquad x>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4db5a7c70c791c69e9f8c4591bd3765fd2cf723c)
![{\displaystyle {d \over dx}c^{x}={c^{x}\ln c},\qquad c>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d595b56138acbd74c00aae36d407fe71f3b2dd7c)
![{\displaystyle {d \over dx}e^{x}=e^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2528d3e349a763e4afd73cddc3ec599ffca15e4)
![{\displaystyle {d \over dx}\log _{c}x={1 \over x\ln c},\qquad c>0,c\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2456ba331f9193749358238ae80a51a579b1bff6)
![{\displaystyle {d \over dx}\ln x={1 \over x},\qquad x>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13505ad13a957d585995644a78568fe7e7b9c398)
![{\displaystyle {d \over dx}\ln |x|={1 \over x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41ee002e46c6ad508619d162d8c484ee5637e265)
![{\displaystyle {d \over dx}x^{x}=x^{x}(1+\ln x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46c2c2e041b4a0a9bcc2549f422a7134def94e97)
![{\displaystyle {d \over dx}{\mbox{ sen }}x=\cos x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b765533ac7735fc71d4a3898ed080110349deb4b)
![{\displaystyle {d \over dx}\cos x=-{\mbox{sen }}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9941021ff2649716377f8ded61471166aa56510c)
![{\displaystyle {d \over dx}{\mbox{ tg }}x=\sec ^{2}x={1 \over \cos ^{2}x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a8b4bf8b61fcf12f63b9e2b317c9442625d185)
![{\displaystyle {d \over dx}\sec x={\mbox{ tg }}x\sec x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3fa0cdbd3e14ece8e2f40dba6d02a76d3771a3a)
![{\displaystyle {d \over dx}{\mbox{ cotg }}x=-{\mbox{cossec }}^{2}x={-1 \over {\mbox{sen}}^{2}x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35143e3b9144d16abb70c2057197b7dffff66ec3)
![{\displaystyle {d \over dx}{\mbox{ cossec }}x=-{\mbox{cossec }}x{\mbox{ cotg }}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d68ce414b4b81bd069357a82e57919be10aa0a9)
![{\displaystyle {d \over dx}{\mbox{ arcsen }}x={1 \over {\sqrt {1-x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8832722e2e1c67e9ff5c89cf14593e8a5bb6448)
![{\displaystyle {d \over dx}\arccos x={-1 \over {\sqrt {1-x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/694f1f1ec9823b1af7a76c435f1be1b8150536a4)
![{\displaystyle {d \over dx}{\mbox{ arctg }}x={1 \over 1+x^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69ee4c1911fee4ab74d9f38f3d80df350432e6c7)
![{\displaystyle {d \over dx}\operatorname {arcsec} x={1 \over |x|{\sqrt {x^{2}-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42da001ad7ea996603263d152a9479be330dafd3)
![{\displaystyle {d \over dx}{\mbox{ arccotg }}x={-1 \over 1+x^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6cced22ffbff76c749f85ab77875d1e20e1500a)
![{\displaystyle {d \over dx}{\mbox{ arccossec }}x={-1 \over |x|{\sqrt {x^{2}-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4356ba199c4e2c98d13d7e9a792bfaade244f488)
![{\displaystyle {d \over dx}{\mbox{ senh }}x=\cosh x={\frac {e^{x}+e^{-x}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/416702d3c273229b44677640ae1674333fb68bdf)
![{\displaystyle {d \over dx}\cosh x={\mbox{ senh }}x={\frac {e^{x}-e^{-x}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffce5bb5e124f9268b2bd84ed872ad7f6ae94a99)
![{\displaystyle {d \over dx}{\mbox{ tgh }}x={\mbox{sech}}^{2}\,x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0381d1551db574222038354f00993fa7f6e4a3)
![{\displaystyle {d \over dx}\,{\mbox{ sech }}\,x=-{\mbox{ tgh }}x\,{\mbox{ sech }}\,x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a35ea7c80c7ec8a82709f38fd3b280948a350a7)
![{\displaystyle {d \over dx}\,{\mbox{ cotgh }}\,x=-\,{\mbox{ cossech}}^{2}\,x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e077b240ae88610aacdec13163677f7769d5b54)
![{\displaystyle {d \over dx}\,\operatorname {csch} \,x=-\,\operatorname {coth} \,x\,\operatorname {cossech} \,x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccba1b336c20da53605769ff5388f0828b1ad15)
![{\displaystyle {d \over dx}\,\operatorname {arcsinh} \,x={1 \over {\sqrt {x^{2}+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/789104eecbf1e0a1862d0a8a54756fd4c92c967a)
![{\displaystyle {d \over dx}\,\operatorname {arccosh} \,x={1 \over {\sqrt {x^{2}-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4c92e415b1105ce4f0520e0c0e8ed59a159a47)
![{\displaystyle {d \over dx}\,\operatorname {arctanh} \,x={1 \over 1-x^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fcf6ba7ee2f290bc1d1ad13386ad085634358cb)
![{\displaystyle {d \over dx}\,\operatorname {arcsech} \,x={-1 \over x{\sqrt {1-x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88e04b2d1c4c3fff12a7878bd16dcd1494ed2d8d)
![{\displaystyle {d \over dx}\,\operatorname {arccoth} \,x={1 \over 1-x^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70f67ea12ad2b5bb503f08fb71d1a0879adb6ce3)
![{\displaystyle {d \over dx}\,\operatorname {arccossech} \,x={-1 \over |x|{\sqrt {1+x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5fe24ce08197a20a0506963b46e8a5586d1c74)
O ALBERTO É O MAIOR!