derivative f(x)=6^xlog_{5}(x)

解

導関数

f (x) = 6^{x} lo g_{5} (x)

\frac{6 ^{x} + ln ( 6 ) \cdot 6 ^{x} x ln ( x )}{ln ( 5 ) x}

解答ステップ

\frac{d}{d x} (6^{x} lo g_{5} (x))

積の法則を適用:

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

f = 6^{x}, g = lo g_{5} (x)

= \frac{d}{d x} (6^{x}) lo g_{5} (x) + \frac{d}{d x} (lo g_{5} (x)) 6^{x}

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

\frac{d}{d x} (lo g_{5} (x)) = \frac{1}{x ln ( 5 )}

= 6^{x} ln (6) lo g_{5} (x) + \frac{1}{x ln ( 5 )} \cdot 6^{x}

簡素化

6^{x} ln (6) lo g_{5} (x) + \frac{1}{x ln ( 5 )} \cdot 6^{x} : \frac{6 ^{x} + ln ( 6 ) \cdot 6 ^{x} x ln ( x )}{ln ( 5 ) x}

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