derivative f(x)=x^4sqrt(sec(5x))

解

導関数

f (x) = x^{4} sec (5 x)

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

解答ステップ

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

積の法則を適用:

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

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

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

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

\frac{d}{d x} (sec (5 x)) = \frac{5 tan ( 5 x ) sec ( 5 x )}{2}

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

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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