derivative of ax*ln(ax^2+3)

Lösung

\frac{d}{d x} (a x \cdot ln (a x^{2} + 3))

a (ln (a x^{2} + 3) + \frac{2 a x ^{2}}{a x ^{2} + 3})

Schritte zur Lösung

\frac{d}{d x} (a x ln (a x^{2} + 3))

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= a \frac{d}{d x} (x ln (a x^{2} + 3))

Wende die Produktregel an:

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

f = x, g = ln (a x^{2} + 3)

= a (\frac{d x}{d x} ln (a x^{2} + 3) + \frac{d}{d x} (ln (a x^{2} + 3)) x)

\frac{d x}{d x} = 1

\frac{d}{d x} (ln (a x^{2} + 3)) = \frac{2 a x}{a x ^{2} + 3}

= a (1 \cdot ln (a x^{2} + 3) + \frac{2 a x}{a x ^{2} + 3} x)

Vereinfache

a (1 \cdot ln (a x^{2} + 3) + \frac{2 a x}{a x ^{2} + 3} x) : a (ln (a x^{2} + 3) + \frac{2 a x ^{2}}{a x ^{2} + 3})

= a (ln (a x^{2} + 3) + \frac{2 a x ^{2}}{a x ^{2} + 3})

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