derivative of e^x(x^2+3(x^3+4))

Lösung

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

e^{x} x^{5} + 5 e^{x} x^{4} + 3 e^{x} x^{3} + 13 e^{x} x^{2} + 8 e^{x} x + 12 e^{x}

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = e^{x}, g = (x^{2} + 3) (x^{3} + 4)

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

\frac{d}{d x} (e^{x}) = e^{x}

\frac{d}{d x} ((x^{2} + 3) (x^{3} + 4)) = 5 x^{4} + 9 x^{2} + 8 x

= e^{x} (x^{2} + 3) (x^{3} + 4) + (5 x^{4} + 9 x^{2} + 8 x) e^{x}

Vereinfache

e^{x} (x^{2} + 3) (x^{3} + 4) + (5 x^{4} + 9 x^{2} + 8 x) e^{x} : e^{x} x^{5} + 5 e^{x} x^{4} + 3 e^{x} x^{3} + 13 e^{x} x^{2} + 8 e^{x} x + 12 e^{x}

= e^{x} x^{5} + 5 e^{x} x^{4} + 3 e^{x} x^{3} + 13 e^{x} x^{2} + 8 e^{x} x + 12 e^{x}

y^{^{''}} + y^{^{'}} = 0

\int \frac{sin ( ln ( 4 ) x ^{2} )}{x} d x

y^{^{'}} = t y

\frac{\partial}{\partial x} ((ln (1 + e^{(x)}))^{2})

t \to 0 lim (\frac{2 e ^{t} - 2}{t})