derivative of 3*4^{x^2+2}

解

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

3 ln (2) \cdot 2^{2 x^{2} + 6} x

解答ステップ

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

定数を除去:

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

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

指数の規則を適用する:

a^{b} = e^{b l n (a)}

4^{x^{2} + 2} = e^{(x^{2} + 2) l n (4)}

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

連鎖法則を適用:

e^{(x^{2} + 2) l n (4)} \frac{d}{d x} ((x^{2} + 2) ln (4))

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

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

= 3 e^{(x^{2} + 2) l n (4)} \cdot 4 ln (2) x

簡素化

3 e^{(x^{2} + 2) l n (4)} \cdot 4 ln (2) x : 3 ln (2) \cdot 2^{2 x^{2} + 6} x

= 3 ln (2) \cdot 2^{2 x^{2} + 6} x