derivative 5x^2sqrt(25-x^2)

Lösung

ableitung von

5 x^{2} 25 - x^{2}

\frac{5 ( - 3 x ^{3} + 50 x )}{25 - x ^{2}}

Schritte zur Lösung

\frac{d}{d x} (5 x^{2} 25 - x^{2})

Entferne die Konstante:

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

= 5 \frac{d}{d x} (x^{2} 25 - x^{2})

Wende die Produktregel an:

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

f = x^{2}, g = 25 - x^{2}

= 5 (\frac{d}{d x} (x^{2}) 25 - x^{2} + \frac{d}{d x} (25 - x^{2}) x^{2})

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

\frac{d}{d x} (25 - x^{2}) = - \frac{x}{25 - x ^{2}}

= 5 (2 x 25 - x^{2} + (- \frac{x}{25 - x ^{2}}) x^{2})

Vereinfache

5 (2 x 25 - x^{2} + (- \frac{x}{25 - x ^{2}}) x^{2}) : \frac{5 ( - 3 x ^{3} + 50 x )}{25 - x ^{2}}

= \frac{5 ( - 3 x ^{3} + 50 x )}{25 - x ^{2}}

\int_{0}^{5} (5 + 3 x)^{- 1} d x

x \to 9 - lim (\frac{x - 9}{x - 9})

\int x \cdot (2 x^{2} + 1)^{\frac{3}{2}} d x

\frac{d x}{d t} = 4 x (9 - x)

\int x^{3} x^{2} + 12 d x