derivative of (e^{(x^2}+4)^2(4x)^3x)

解

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

64 (4 x^{3} (e^{x^{2}} + 4)^{2} + 4 e^{x^{2}} x^{5} (e^{x^{2}} + 4))

解答ステップ

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

簡素化

(e^{(x^{2})} + 4)^{2} (4 x)^{3} x : 64 x^{4} (e^{x^{2}} + 4)^{2}

= \frac{d}{d x} (64 x^{4} (e^{x^{2}} + 4)^{2})

定数を除去:

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

= 64 \frac{d}{d x} (x^{4} (e^{x^{2}} + 4)^{2})

積の法則を適用:

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

f = x^{4}, g = (e^{x^{2}} + 4)^{2}

= 64 (\frac{d}{d x} (x^{4}) (e^{x^{2}} + 4)^{2} + \frac{d}{d x} ((e^{x^{2}} + 4)^{2}) x^{4})

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

\frac{d}{d x} ((e^{x^{2}} + 4)^{2}) = 4 e^{x^{2}} x (e^{x^{2}} + 4)

= 64 (4 x^{3} (e^{x^{2}} + 4)^{2} + 4 e^{x^{2}} x (e^{x^{2}} + 4) x^{4})

4 e^{x^{2}} x (e^{x^{2}} + 4) x^{4} = 4 e^{x^{2}} x^{5} (e^{x^{2}} + 4)

= 64 (4 x^{3} (e^{x^{2}} + 4)^{2} + 4 e^{x^{2}} x^{5} (e^{x^{2}} + 4))