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

解

\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)

解答ステップ

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

定数を除去:

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

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

積の法則を適用:

(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))

簡素化

\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)