derivative of (x^3+x^2+x+5ln(5x^2-7))

Lösung

\frac{d}{d x} ((x^{3} + x^{2} + x + 5) ln (5 x^{2} - 7))

(3 x^{2} + 2 x + 1) ln (5 x^{2} - 7) + \frac{10 x}{5 x ^{2} - 7} (x^{3} + x^{2} + x + 5)

Schritte zur Lösung

\frac{d}{d x} ((x^{3} + x^{2} + x + 5) ln (5 x^{2} - 7))

Wende die Produktregel an:

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

f = x^{3} + x^{2} + x + 5, g = ln (5 x^{2} - 7)

= \frac{d}{d x} (x^{3} + x^{2} + x + 5) ln (5 x^{2} - 7) + \frac{d}{d x} (ln (5 x^{2} - 7)) (x^{3} + x^{2} + x + 5)

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

\frac{d}{d x} (ln (5 x^{2} - 7)) = \frac{10 x}{5 x ^{2} - 7}

= (3 x^{2} + 2 x + 1) ln (5 x^{2} - 7) + \frac{10 x}{5 x ^{2} - 7} (x^{3} + x^{2} + x + 5)