derivative x^3sec(x)+sec(x)tan(x)x^4

解

導関数

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

x^{4} tan^{2} (x) sec (x) + x^{4} sec^{3} (x) + 5 x^{3} sec (x) tan (x) + 3 x^{2} sec (x)

解答ステップ

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

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

\frac{d}{d x} (x^{3} sec (x)) = 3 x^{2} sec (x) + sec (x) tan (x) x^{3}

\frac{d}{d x} (sec (x) tan (x) x^{4}) = x^{4} tan^{2} (x) sec (x) + sec (x) (x^{4} sec^{2} (x) + 4 x^{3} tan (x))

= 3 x^{2} sec (x) + sec (x) tan (x) x^{3} + x^{4} tan^{2} (x) sec (x) + sec (x) (x^{4} sec^{2} (x) + 4 x^{3} tan (x))

簡素化

3 x^{2} sec (x) + sec (x) tan (x) x^{3} + x^{4} tan^{2} (x) sec (x) + sec (x) (x^{4} sec^{2} (x) + 4 x^{3} tan (x)) : x^{4} tan^{2} (x) sec (x) + x^{4} sec^{3} (x) + 5 x^{3} sec (x) tan (x) + 3 x^{2} sec (x)

= x^{4} tan^{2} (x) sec (x) + x^{4} sec^{3} (x) + 5 x^{3} sec (x) tan (x) + 3 x^{2} sec (x)