derivative of (3x^4/(tan(x)))

解

\frac{d}{d x} (\frac{3 x ^{4}}{tan ( x )})

3 (4 x^{3} cot (x) - x^{4} csc^{2} (x))

解答ステップ

\frac{d}{d x} (\frac{3 x ^{4}}{tan ( x )})

簡素化

\frac{3 x ^{4}}{tan ( x )} : 3 x^{4} cot (x)

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

定数を除去:

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

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

積の法則を適用:

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

f = x^{4}, g = cot (x)

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

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

\frac{d}{d x} (cot (x)) = - csc^{2} (x)

= 3 (4 x^{3} cot (x) + (- csc^{2} (x)) x^{4})

簡素化

= 3 (4 x^{3} cot (x) - x^{4} csc^{2} (x))