derivative of 3^xlog_{5}(x)

解

\frac{d}{d x} (3^{x} lo g_{5} (x))

\frac{3 ^{x} + ln ( 3 ) \cdot 3 ^{x} x ln ( x )}{ln ( 5 ) x}

解答ステップ

\frac{d}{d x} (3^{x} lo g_{5} (x))

積の法則を適用:

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

f = 3^{x}, g = lo g_{5} (x)

= \frac{d}{d x} (3^{x}) lo g_{5} (x) + \frac{d}{d x} (lo g_{5} (x)) 3^{x}

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

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

= 3^{x} ln (3) lo g_{5} (x) + \frac{1}{x ln ( 5 )} \cdot 3^{x}

簡素化

3^{x} ln (3) lo g_{5} (x) + \frac{1}{x ln ( 5 )} \cdot 3^{x} : \frac{3 ^{x} + ln ( 3 ) \cdot 3 ^{x} x ln ( x )}{ln ( 5 ) x}

= \frac{3 ^{x} + ln ( 3 ) \cdot 3 ^{x} x ln ( x )}{ln ( 5 ) x}