derivative of (x^2+5x+1{f}(x)(sin(3x)))

解答

\frac{d}{d x} ((x^{2} + 5 x + 1) f (x) (sin (3 x)))

(2 x + 5) f (x) sin (3 x) + (\frac{d}{d x} (f (x)) sin (3 x) + cos (3 x) \cdot 3 f (x)) (x^{2} + 5 x + 1)

求解步骤

\frac{d}{d x} ((x^{2} + 5 x + 1) f (x) (sin (3 x)))

使用微分乘法定则:

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

f = x^{2} + 5 x + 1, g = f (x) sin (3 x)

= \frac{d}{d x} (x^{2} + 5 x + 1) f (x) sin (3 x) + \frac{d}{d x} (f (x) sin (3 x)) (x^{2} + 5 x + 1)

\frac{d}{d x} (x^{2} + 5 x + 1) = 2 x + 5

\frac{d}{d x} (f (x) sin (3 x)) = \frac{d}{d x} (f (x)) sin (3 x) + cos (3 x) \cdot 3 f (x)

= (2 x + 5) f (x) sin (3 x) + (\frac{d}{d x} (f (x)) sin (3 x) + cos (3 x) \cdot 3 f (x)) (x^{2} + 5 x + 1)