derivative of ln(2x^4e^{3x}(x^2+x+1^5))

Lösung

\frac{d}{d x} (ln (2 x^{4} e^{3 x} (x^{2} + x + 1)^{5}))

\frac{14 x ^{2} + 9 x + 4}{x ( x ^{2} + x + 1 )} + 3

Schritte zur Lösung

\frac{d}{d x} (ln (2 x^{4} e^{3 x} (x^{2} + x + 1)^{5}))

Vereinfache

ln (2 x^{4} e^{3 x} (x^{2} + x + 1)^{5}) : ln (2 x^{4} (x^{2} + x + 1)^{5}) + 3 x

= \frac{d}{d x} (ln (2 x^{4} (x^{2} + x + 1)^{5}) + 3 x)

Wende die Summen-/Differenzregel an:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (ln (2 x^{4} (x^{2} + x + 1)^{5})) + \frac{d}{d x} (3 x)

\frac{d}{d x} (ln (2 x^{4} (x^{2} + x + 1)^{5})) = \frac{14 x ^{2} + 9 x + 4}{x ( x ^{2} + x + 1 )}

\frac{d}{d x} (3 x) = 3

= \frac{14 x ^{2} + 9 x + 4}{x ( x ^{2} + x + 1 )} + 3

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