derivative of e^{-3x^4}*ln(7x^2)

Lösung

\frac{d}{d x} (e^{- 3 x^{4}} \cdot ln (7 x^{2}))

- 12 e^{- 3 x^{4}} x^{3} ln (7 x^{2}) + \frac{2 e ^{- 3 x^{4}}}{x}

Schritte zur Lösung

\frac{d}{d x} (e^{- 3 x^{4}} ln (7 x^{2}))

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{- 3 x^{4}}, g = ln (7 x^{2})

= \frac{d}{d x} (e^{- 3 x^{4}}) ln (7 x^{2}) + \frac{d}{d x} (ln (7 x^{2})) e^{- 3 x^{4}}

\frac{d}{d x} (e^{- 3 x^{4}}) = - 12 e^{- 3 x^{4}} x^{3}

\frac{d}{d x} (ln (7 x^{2})) = \frac{2}{x}

= (- 12 e^{- 3 x^{4}} x^{3}) ln (7 x^{2}) + \frac{2}{x} e^{- 3 x^{4}}

Vereinfache

(- 12 e^{- 3 x^{4}} x^{3}) ln (7 x^{2}) + \frac{2}{x} e^{- 3 x^{4}} : - 12 e^{- 3 x^{4}} x^{3} ln (7 x^{2}) + \frac{2 e ^{- 3 x^{4}}}{x}

= - 12 e^{- 3 x^{4}} x^{3} ln (7 x^{2}) + \frac{2 e ^{- 3 x^{4}}}{x}

\frac{\partial}{\partial y} (\frac{x}{y ^{2}})

x \to 0 lim (\frac{7}{1 + x})

\frac{d}{d x} (3 x^{2} - x^{4})

\int (\frac{2 x ^{3} + 10 x ^{2} + 8 x - 12}{4 x}) d x

\int \frac{x}{x ^{2} + 9} d x