derivative of (4csc(x)/(5^x))

解

\frac{d}{d x} (\frac{4 csc ( x )}{5 ^{x}})

- \frac{4 csc ( x ) ( cot ( x ) + ln ( 5 ) )}{5 ^{x}}

解答ステップ

\frac{d}{d x} (\frac{4 csc ( x )}{5 ^{x}})

定数を除去:

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

= 4 \frac{d}{d x} (\frac{csc ( x )}{5 ^{x}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= 4 \cdot \frac{\frac{d}{d x} ( csc ( x ) ) 5 ^{x} - \frac{d}{d x} ( 5 ^{x} ) csc ( x )}{( 5 ^{x} ) ^{2}}

\frac{d}{d x} (csc (x)) = - cot (x) csc (x)

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

= 4 \cdot \frac{( - cot ( x ) csc ( x ) ) \cdot 5 ^{x} - 5 ^{x} ln ( 5 ) csc ( x )}{( 5 ^{x} ) ^{2}}

簡素化

4 \cdot \frac{( - cot ( x ) csc ( x ) ) \cdot 5 ^{x} - 5 ^{x} ln ( 5 ) csc ( x )}{( 5 ^{x} ) ^{2}} : - \frac{4 csc ( x ) ( cot ( x ) + ln ( 5 ) )}{5 ^{x}}

= - \frac{4 csc ( x ) ( cot ( x ) + ln ( 5 ) )}{5 ^{x}}