A operação primária do cálculo diferencial é encontrar a derivada de uma função. Na tabela a seguir[1], supomos que f {\displaystyle f} e g {\displaystyle g} são funções deriváveis em R {\displaystyle \mathbb {R} } e c {\displaystyle c} é um número real. Essas fórmulas são suficientes para derivar qualquer função elementar. Demonstrações destas fórmulas podem ser obtidas em livros de cálculo diferencial e integral[2][3][4][5]. Regra da soma ( f + g ) ′ = f ′ + g ′ {\displaystyle \left({f+g}\right)'=f'+g'} Regra da subtração ( f − g ) ′ = f ′ − g ′ {\displaystyle (f-g)'=f'-g'} Regra da multiplicação ( c f ) ′ = c f ′ {\displaystyle (cf)'=cf'} Regra do produto ( f g ) ′ = f ′ g + f g ′ {\displaystyle \left({fg}\right)'=f'g+fg'} Regra do quociente ( f g ) ′ = f ′ g − f g ′ g 2 {\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-fg'}{g^{2}}}} sendo esta válida para todo x {\displaystyle x} no domínio das funções com g ( x ) ≠ 0 {\displaystyle g(x)\neq 0} . Regra da Cadeia ( f ∘ g ) ′ ( x ) = f ′ ( g ( x ) ) g ′ ( x ) {\displaystyle (f\circ g)'(x)=f'{\big (}g(x){\big )}g'(x)} onde ( f ∘ g ) ( x ) := f ( g ( x ) ) {\displaystyle (f\circ g)(x):=f{\big (}g(x){\big )}} é a composição de f {\displaystyle f} com g {\displaystyle g} (usualmente, lê-se " f {\displaystyle f} após g {\displaystyle g} "). Esta é válida para x {\displaystyle x} no domínio D g {\displaystyle D_{g}} da função g {\displaystyle g} e tal que g ( x ) {\displaystyle g(x)} esteja no domínio D f {\displaystyle D_{f}} da função f {\displaystyle f} , ou seja, é válida em D f ∘ g = { x ∈ D g : g ( x ) ∈ D f } {\displaystyle D_{f\circ g}=\{x\in D_{g}:g(x)\in D_{f}\}} . d d x c = 0 {\displaystyle {d \over dx}c=0} d d x x = 1 {\displaystyle {d \over dx}x=1} d d x c . x = c {\displaystyle {d \over dx}c.x=c} d d x x c = c x c − 1 {\displaystyle {d \over dx}x^{c}=cx^{c-1}} d d x c x = c x ln c , c > 0 {\displaystyle {\frac {d}{dx}}c^{x}=c^{x}\ln c,\quad c>0} d d x e x = e x {\displaystyle {\frac {d}{dx}}e^{x}=e^{x}} d d x log b | x | = 1 x ln b {\displaystyle {\frac {d}{dx}}\log _{b}|x|={\frac {1}{x\ln b}}} d d x ln | x | = 1 x {\displaystyle {\frac {d}{dx}}\ln |x|={\frac {1}{x}}} Se u {\displaystyle u} é uma função derivável, então: d d x e u = u ′ e u {\displaystyle {\frac {d}{dx}}e^{u}=u'e^{u}} d d x a u = u ′ a u ln a {\displaystyle {\frac {d}{dx}}a^{u}=u'a^{u}\ln a} d d x log a u = u ′ u log a e {\displaystyle {\frac {d}{dx}}\log _{a}u={\frac {u'}{u}}\log _{a}e} d d x ln | u | = u ′ u {\displaystyle {\frac {d}{dx}}\ln |u|={\frac {u'}{u}}} Mais informação , ... Função Abreviatura Identidade trigonométrica Seno sen (ou sin) sen θ ≡ cos ( π 2 − θ ) ≡ 1 csc θ {\displaystyle \operatorname {sen} \theta \equiv \cos \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\csc \theta }}} Cosseno cos cos θ ≡ sen ( π 2 − θ ) ≡ 1 sec θ {\displaystyle \cos \theta \equiv \operatorname {sen} \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\sec \theta }}} Tangente tan (ou tg) tan θ ≡ sen θ cos θ ≡ cot ( π 2 − θ ) ≡ 1 cot θ {\displaystyle \tan \theta \equiv {\frac {\operatorname {sen} \theta }{\cos \theta }}\equiv \cot \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\cot \theta }}} Cossecante csc (ou cosec) csc θ ≡ sec ( π 2 − θ ) ≡ 1 sen θ {\displaystyle \csc \theta \equiv \sec \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\operatorname {sen} \theta }}} Secante sec sec θ ≡ csc ( π 2 − θ ) ≡ 1 cos θ {\displaystyle \sec \theta \equiv \csc \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\cos \theta }}} Cotangente cot (ou cotg ou cotan) cot θ ≡ cos θ sen θ ≡ tan ( π 2 − θ ) ≡ 1 tan θ {\displaystyle \cot \theta \equiv {\frac {\cos \theta }{\operatorname {sen} \theta }}\equiv \tan \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {1}{\tan \theta }}} Fechar d d x sen x = cos x {\displaystyle {\frac {d}{dx}}\operatorname {sen} x=\cos x} d d x cos x = − sen x {\displaystyle {\frac {d}{dx}}\cos x=-\operatorname {sen} x} d d x tan x = sec 2 x {\displaystyle {\frac {d}{dx}}\tan x=\sec ^{2}x} d d x csc x = − csc x cot x {\displaystyle {d \over dx}\csc x=-\csc x\cot x} d d x sec x = sec x tan x {\displaystyle {d \over dx}\sec x=\sec x\tan x} d d x cot x = − csc 2 x {\displaystyle {d \over dx}\cot x=-\csc ^{2}x} d d x arcsen x = 1 1 − x 2 {\displaystyle {d \over dx}\operatorname {arcsen} x={\frac {1}{\sqrt {1-x^{2}}}}} d d x arccos x = − 1 1 − x 2 {\displaystyle {d \over dx}\arccos x=-{\frac {1}{\sqrt {1-x^{2}}}}} d d x arctan x = 1 1 + x 2 {\displaystyle {d \over dx}\arctan \,x={\frac {1}{1+x^{2}}}} d d x arcsec x = 1 | x | x 2 − 1 {\displaystyle {d \over dx}\operatorname {arcsec} x={\frac {1}{|x|{\sqrt {x^{2}-1}}}}} d d x arccot x = − 1 1 + x 2 {\displaystyle {d \over dx}\operatorname {arccot} x=-{\frac {1}{1+x^{2}}}} d d x arccsc x = − 1 | x | x 2 − 1 {\displaystyle {d \over dx}\operatorname {arccsc} x=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}} d d x senh x = cosh x {\displaystyle {d \over dx}\operatorname {senh} x=\cosh x} d d x cosh x = senh x {\displaystyle {d \over dx}\cosh x=\operatorname {senh} x} d d x tanh x = sech 2 x {\displaystyle {d \over dx}\tanh x=\operatorname {sech} ^{2}x} d d x sech x = − sech x tanh x {\displaystyle {d \over dx}\operatorname {sech} x=-\operatorname {sech} x\tanh x} d d x coth x = − csch 2 x {\displaystyle {d \over dx}\operatorname {coth} x=-\operatorname {csch} ^{2}x} d d x csch x = − csch x coth x {\displaystyle {d \over dx}\,\operatorname {csch} x=-\operatorname {csch} x\operatorname {coth} x} d d x argsenh x = 1 x 2 + 1 {\displaystyle {d \over dx}\operatorname {argsenh} x={\frac {1}{\sqrt {x^{2}+1}}}} d d x argcosh x = 1 x 2 − 1 {\displaystyle {d \over dx}\operatorname {argcosh} x={\frac {1}{\sqrt {x^{2}-1}}}} d d x argtanh x = 1 1 − x 2 {\displaystyle {d \over dx}\operatorname {argtanh} x={\frac {1}{1-x^{2}}}} d d x argsech x = − 1 x 1 − x 2 {\displaystyle {d \over dx}\operatorname {argsech} x=-{\frac {1}{x{\sqrt {1-x^{2}}}}}} d d x argcoth x = 1 1 − x 2 {\displaystyle {d \over dx}\operatorname {argcoth} x={\frac {1}{1-x^{2}}}} d d x argcsch x = − 1 | x | x 2 + 1 {\displaystyle {d \over dx}\operatorname {argcsch} x=-{1 \over |x|{\sqrt {x^{2}+1}}}} Tabela de integrais [1]«Faça exemplos com O Monitor». omonitor.io. Consultado em 22 de março de 2016 [2]Leithold, Louis (1994). Cálculo com geometria analítica - vol. 1 3. ed. [S.l.]: Harbra. ISBN 8529400941 [3]Simmons, George (2009). Calculo com geometria analitica. [S.l.]: Pearson Makron Books. ISBN 0074504118 [4]Howard, Anton (2007). Cálculo - vol 1. 8 ed. [S.l.]: Bookman. ISBN 9788560031634 [5]Stewart, James (2006). Cálculo - Vol. 1 5 ed. [S.l.]: Thompson. ISBN 8522104794 Portal da matemática Wikiwand in your browser!Seamless Wikipedia browsing. On steroids.Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.Wikiwand for ChromeWikiwand for Firefox
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