derivative of 1/3 pi{r}(x(x)^2h)

Lösung

\frac{d}{d x} (\frac{1}{3} π r (x) (x)^{2} h)

\frac{πh ( x ^{2} \frac{d}{d x} ( r ( x ) ) + 2 x r ( x ) )}{3}

Schritte zur Lösung

\frac{d}{d x} (\frac{1}{3} π r (x) (x)^{2} h)

Entferne die Konstante:

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

= \frac{1}{3} πh \frac{d}{d x} (r (x) x^{2})

Wende die Produktregel an:

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

f = r (x), g = x^{2}

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

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

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

Vereinfache

\frac{1}{3} πh (\frac{d}{d x} (r (x)) x^{2} + 2 x r (x)) : \frac{πh ( x ^{2} \frac{d}{d x} ( r ( x ) ) + 2 x r ( x ) )}{3}

= \frac{πh ( x ^{2} \frac{d}{d x} ( r ( x ) ) + 2 x r ( x ) )}{3}

\int_{0}^{7} \frac{1}{9 + x ^{2}} d x

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