derivative of 4/3 pi{r}(x(x)^3)

Lösung

\frac{d}{d x} (\frac{4}{3} π r (x) (x)^{3})

\frac{4 π x ^{3} \frac{d}{d x} ( r ( x ) )}{3} + 4 π x^{2} r (x)

Schritte zur Lösung

\frac{d}{d x} (\frac{4}{3} π r (x) (x)^{3})

Entferne die Konstante:

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

= \frac{4}{3} π \frac{d}{d x} (r (x) x^{3})

Wende die Produktregel an:

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

f = r (x), g = x^{3}

= \frac{4}{3} π (\frac{d}{d x} (r (x)) x^{3} + \frac{d}{d x} (x^{3}) r (x))

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

= \frac{4}{3} π (\frac{d}{d x} (r (x)) x^{3} + 3 x^{2} r (x))

Vereinfache

\frac{4}{3} π (\frac{d}{d x} (r (x)) x^{3} + 3 x^{2} r (x)) : \frac{4 π x ^{3} \frac{d}{d x} ( r ( x ) )}{3} + 4 π x^{2} r (x)

= \frac{4 π x ^{3} \frac{d}{d x} ( r ( x ) )}{3} + 4 π x^{2} r (x)

\frac{d}{d x} (\frac{- x ^{2} - 8}{( x ^{2} - 8 ) ^{2}})

\frac{d x}{d t} = - \frac{6 x}{20 + 2 t}

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

a re a y = x^{3} - 3 x^{2} - x + 3, - 1 < x \leq 3

x \to 1 + lim (\frac{1}{2} x + 3)