derivative f(x)=x^2ln(10-5x^2)

Lösung

ableitung von

f (x) = x^{2} ln (10 - 5 x^{2})

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = x^{2}, g = ln (10 - 5 x^{2})

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

\frac{d}{d x} (x^{2}) = 2 x

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

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

Vereinfache

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

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

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