derivative of 5sin(x(4/(x^3)+3x))

解答

\frac{d}{d x} (5 sin (x) (\frac{4}{x ^{3}} + 3 x))

5 (cos (x) (\frac{4}{x ^{3}} + 3 x) + (- \frac{12}{x ^{4}} + 3) sin (x))

求解步骤

\frac{d}{d x} (5 sin (x) (\frac{4}{x ^{3}} + 3 x))

将常数提出:

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

= 5 \frac{d}{d x} (sin (x) (\frac{4}{x ^{3}} + 3 x))

使用微分乘法定则:

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

f = sin (x), g = \frac{4}{x ^{3}} + 3 x

= 5 (\frac{d}{d x} (sin (x)) (\frac{4}{x ^{3}} + 3 x) + \frac{d}{d x} (\frac{4}{x ^{3}} + 3 x) sin (x))

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

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

= 5 (cos (x) (\frac{4}{x ^{3}} + 3 x) + (- \frac{12}{x ^{4}} + 3) sin (x))

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