derivative of 3x^2cos(x^3)

解答

\frac{d}{d x} (3 x^{2} cos (x^{3}))

3 (2 x cos (x^{3}) - 3 x^{4} sin (x^{3}))

求解步骤

\frac{d}{d x} (3 x^{2} cos (x^{3}))

将常数提出:

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

= 3 \frac{d}{d x} (x^{2} cos (x^{3}))

使用微分乘法定则:

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

f = x^{2}, g = cos (x^{3})

= 3 (\frac{d}{d x} (x^{2}) cos (x^{3}) + \frac{d}{d x} (cos (x^{3})) x^{2})

\frac{d}{d x} (x^{2}) = 2 x

\frac{d}{d x} (cos (x^{3})) = - sin (x^{3}) \cdot 3 x^{2}

= 3 (2 x cos (x^{3}) + (- sin (x^{3}) \cdot 3 x^{2}) x^{2})

化简

3 (2 x cos (x^{3}) + (- sin (x^{3}) \cdot 3 x^{2}) x^{2}) : 3 (2 x cos (x^{3}) - 3 x^{4} sin (x^{3}))

= 3 (2 x cos (x^{3}) - 3 x^{4} sin (x^{3}))

\frac{d y}{d t} + 3 y = cos (2 t)

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