derivative of 5^{6^{7^x}}

解

\frac{d}{d x} (5^{6^{7^{x}}})

ln (5) ln (6) ln (7) \cdot 7^{x} \cdot 5^{6^{7^{x}}} \cdot 6^{7^{x}}

解答ステップ

\frac{d}{d x} (5^{6^{7^{x}}})

指数の規則を適用する:

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

5^{6^{7^{x}}} = e^{6^{7^{x}} l n (5)}

= \frac{d}{d x} (e^{6^{7^{x}} l n (5)})

連鎖法則を適用:

e^{6^{7^{x}} l n (5)} \frac{d}{d x} (6^{7^{x}} ln (5))

= e^{6^{7^{x}} l n (5)} \frac{d}{d x} (6^{7^{x}} ln (5))

\frac{d}{d x} (6^{7^{x}} ln (5)) = ln (5) ln (6) ln (7) \cdot 7^{x} \cdot 6^{7^{x}}

= e^{6^{7^{x}} l n (5)} ln (5) ln (6) ln (7) \cdot 7^{x} \cdot 6^{7^{x}}

e^{6^{7^{x}} l n (5)} = 5^{6^{7^{x}}}

= ln (5) ln (6) ln (7) \cdot 7^{x} \cdot 5^{6^{7^{x}}} \cdot 6^{7^{x}}