derivative of ((x^5+3x^4)/(cos(x)))

解

\frac{d}{d x} (\frac{( x ^{5} + 3 x ) ^{4}}{cos ( x )})

sec (x) tan (x) (x^{5} + 3 x)^{4} + 4 (x^{5} + 3 x)^{3} (5 x^{4} + 3) sec (x)

解答ステップ

\frac{d}{d x} (\frac{( x ^{5} + 3 x ) ^{4}}{cos ( x )})

簡素化

\frac{( x ^{5} + 3 x ) ^{4}}{cos ( x )} : sec (x) (x^{5} + 3 x)^{4}

= \frac{d}{d x} (sec (x) (x^{5} + 3 x)^{4})

積の法則を適用:

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

f = sec (x), g = (x^{5} + 3 x)^{4}

= \frac{d}{d x} (sec (x)) (x^{5} + 3 x)^{4} + \frac{d}{d x} ((x^{5} + 3 x)^{4}) sec (x)

\frac{d}{d x} (sec (x)) = sec (x) tan (x)

\frac{d}{d x} ((x^{5} + 3 x)^{4}) = 4 (x^{5} + 3 x)^{3} (5 x^{4} + 3)

= sec (x) tan (x) (x^{5} + 3 x)^{4} + 4 (x^{5} + 3 x)^{3} (5 x^{4} + 3) sec (x)