derivative of 1/(x^3+4x^2e^{xcos(x)})

Lösung

\frac{d}{d x} (\frac{1}{x ^{3} + 4 x ^{2}} e^{x c o s (x)})

- \frac{e ^{x c o s (x)} ( 3 x + 8 )}{x ^{3} ( x + 4 ) ^{2}} + \frac{e ^{x c o s (x)} ( cos ( x ) - x sin ( x ) )}{x ^{3} + 4 x ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{1}{x ^{3} + 4 x ^{2}} e^{x c o s (x)})

Wende die Produktregel an:

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

f = \frac{1}{x ^{3} + 4 x ^{2}}, g = e^{x c o s (x)}

= \frac{d}{d x} (\frac{1}{x ^{3} + 4 x ^{2}}) e^{x c o s (x)} + \frac{d}{d x} (e^{x c o s (x)}) \frac{1}{x ^{3} + 4 x ^{2}}

\frac{d}{d x} (\frac{1}{x ^{3} + 4 x ^{2}}) = - \frac{3 x + 8}{x ^{3} ( x + 4 ) ^{2}}

\frac{d}{d x} (e^{x c o s (x)}) = e^{x c o s (x)} (cos (x) - x sin (x))

= (- \frac{3 x + 8}{x ^{3} ( x + 4 ) ^{2}}) e^{x c o s (x)} + e^{x c o s (x)} (cos (x) - x sin (x)) \frac{1}{x ^{3} + 4 x ^{2}}

Vereinfache

(- \frac{3 x + 8}{x ^{3} ( x + 4 ) ^{2}}) e^{x c o s (x)} + e^{x c o s (x)} (cos (x) - x sin (x)) \frac{1}{x ^{3} + 4 x ^{2}} : - \frac{e ^{x c o s (x)} ( 3 x + 8 )}{x ^{3} ( x + 4 ) ^{2}} + \frac{e ^{x c o s (x)} ( cos ( x ) - x sin ( x ) )}{x ^{3} + 4 x ^{2}}

= - \frac{e ^{x c o s (x)} ( 3 x + 8 )}{x ^{3} ( x + 4 ) ^{2}} + \frac{e ^{x c o s (x)} ( cos ( x ) - x sin ( x ) )}{x ^{3} + 4 x ^{2}}

t a y l or x e^{- x}

x \to - 5 lim (x - 5)

a re a X^{2} - 2 X, (0, 3)

\frac{d}{d x} (\frac{1 - 3 x}{6 + x})

B^{^{'}}