derivative of (3x^4-18x^5*cos(5x^2))

解

\frac{d}{d x} ((3 x^{4} - 18 x)^{5} \cdot cos (5 x^{2}))

5 cos (5 x^{2}) (3 x^{4} - 18 x)^{4} (12 x^{3} - 18) - 10 x sin (5 x^{2}) (3 x^{4} - 18 x)^{5}

解答ステップ

\frac{d}{d x} ((3 x^{4} - 18 x)^{5} cos (5 x^{2}))

積の法則を適用:

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

f = (3 x^{4} - 18 x)^{5}, g = cos (5 x^{2})

= \frac{d}{d x} ((3 x^{4} - 18 x)^{5}) cos (5 x^{2}) + \frac{d}{d x} (cos (5 x^{2})) (3 x^{4} - 18 x)^{5}

\frac{d}{d x} ((3 x^{4} - 18 x)^{5}) = 5 (3 x^{4} - 18 x)^{4} (12 x^{3} - 18)

\frac{d}{d x} (cos (5 x^{2})) = - sin (5 x^{2}) \cdot 10 x

= 5 (3 x^{4} - 18 x)^{4} (12 x^{3} - 18) cos (5 x^{2}) + (- sin (5 x^{2}) \cdot 10 x) (3 x^{4} - 18 x)^{5}

簡素化

= 5 cos (5 x^{2}) (3 x^{4} - 18 x)^{4} (12 x^{3} - 18) - 10 x sin (5 x^{2}) (3 x^{4} - 18 x)^{5}