derivative f(x)=5x(sin(x)+cos(x))

解

導関数

f (x) = 5 x (sin (x) + cos (x))

5 (sin (x) + cos (x) + x (cos (x) - sin (x)))

解答ステップ

\frac{d}{d x} (5 x (sin (x) + cos (x)))

定数を除去:

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

= 5 \frac{d}{d x} (x (sin (x) + cos (x)))

積の法則を適用:

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

f = x, g = sin (x) + cos (x)

= 5 (\frac{d x}{d x} (sin (x) + cos (x)) + \frac{d}{d x} (sin (x) + cos (x)) x)

\frac{d x}{d x} = 1

\frac{d}{d x} (sin (x) + cos (x)) = cos (x) - sin (x)

= 5 (1 \cdot (sin (x) + cos (x)) + (cos (x) - sin (x)) x)

簡素化

= 5 (sin (x) + cos (x) + x (cos (x) - sin (x)))