Derivadas
Regras de Derivadas
Regra da Potência
\(\frac{d}{dx} x^a = a \cdot x^{a-1}\)
Derivada de uma Constante
\(\frac{d}{dx} a = 0\)
Regra da Soma e da Diferença
\((f \pm g)' = f' \pm g'\)
Constante Fora
\((a \cdot f)' = a \cdot f'\)
Regra do Produto
\((f \cdot g)' = f' \cdot g + f \cdot g'\)
Regra do Quociente
\(\left(\frac{f}{g}\right)' = \frac{f' \cdot g - g' \cdot f}{g^2}\)
Regra da Cadeia
\(\frac{df(u)}{dx} = \frac{df}{du} \cdot \frac{du}{dx}\)
Derivadas Comuns
\(\frac{d}{dx} \ln|x| = \frac{1}{x}\)
\(\frac{d}{dx} \ln x = \frac{1}{x}\)
\(\frac{d}{dx} e^x = e^x\)
\(\frac{d}{dx} \log x = \frac{1}{x \ln 10}\)
\(\frac{d}{dx} \log_a x = \frac{1}{x \ln a}\)
Derivadas Trigonométricas
\(\frac{d}{dx} \sin x = \cos x\)
\(\frac{d}{dx} \cos x = -\sin x\)
\(\frac{d}{dx} \tan x = \sec^2 x\)
\(\frac{d}{dx} \sec x = \frac{\tan x}{\cos^2 x}\)
\(\frac{d}{dx} \csc x = \frac{-\cot x}{\sin^2 x}\)
\(\frac{d}{dx} \cot x = \frac{-1}{\sin^2 x}\)
Derivadas Arc Trigonométricas
\(\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}}\)
\(\frac{d}{dx} \arccos x = \frac{-1}{\sqrt{1 - x^2}}\)
\(\frac{d}{dx} \arctan x = \frac{1}{x^2 + 1}\)
\(\frac{d}{dx} \text{arcsec } x = \frac{1}{|x|\sqrt{x^2 - 1}}\)
\(\frac{d}{dx} \text{arccsc } x = \frac{-1}{|x|\sqrt{x^2 - 1}}\)
\(\frac{d}{dx} \text{arccot } x = \frac{-1}{x^2 + 1}\)
Derivadas Hiperbólicas
\(\frac{d}{dx} \sinh x = \cosh x\)
\(\frac{d}{dx} \cosh x = \sinh x\)
\(\frac{d}{dx} \tanh x = \text{sech}^2 x\)
\(\frac{d}{dx} \text{sech } x = -\tanh x \cdot \text{sech } x\)
\(\frac{d}{dx} \text{csch } x = -\coth x \cdot \text{csch } x\)
\(\frac{d}{dx} \coth x = -\text{csch}^2 x\)
Derivadas Arc Hiperbólicas
\(\frac{d}{dx} \text{arcsinh } x = \frac{1}{\sqrt{x^2 + 1}}\)
\(\frac{d}{dx} \text{arccosh } x = \frac{1}{\sqrt{x - 1}\sqrt{x + 1}}\)
\(\frac{d}{dx} \text{arctanh } x = \frac{1}{1 - x^2}\)
\(\frac{d}{dx} \text{arcsech } x = \frac{-1}{x\sqrt{\frac{x - 1}{x + 1}} \cdot 2 - 1}\)
\(\frac{d}{dx} \text{arccsch } x = \frac{-1}{|x|\sqrt{1 + \frac{1}{x^2}}}\)
\(\frac{d}{dx} \text{arccoth } x = \frac{1}{1 - x^2}\)