derivative (3x^6)/(cos(x)e^x)

解答

导数

\frac{3 x ^{6}}{cos ( x ) e ^{x}}

\frac{3 x ^{5} ( 6 cos ( x ) - x ( cos ( x ) - sin ( x ) ) )}{e ^{x} cos ^{2} ( x )}

求解步骤

\frac{d}{d x} (\frac{3 x ^{6}}{cos ( x ) e ^{x}})

将常数提出:

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

= 3 \frac{d}{d x} (\frac{x ^{6}}{cos ( x ) e ^{x}})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= 3 \cdot \frac{\frac{d}{d x} ( x ^{6} ) cos ( x ) e ^{x} - \frac{d}{d x} ( cos ( x ) e ^{x} ) x ^{6}}{( cos ( x ) e ^{x} ) ^{2}}

\frac{d}{d x} (x^{6}) = 6 x^{5}

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

= 3 \cdot \frac{6 x ^{5} cos ( x ) e ^{x} - ( - e ^{x} sin ( x ) + e ^{x} cos ( x ) ) x ^{6}}{( cos ( x ) e ^{x} ) ^{2}}

化简

3 \cdot \frac{6 x ^{5} cos ( x ) e ^{x} - ( - e ^{x} sin ( x ) + e ^{x} cos ( x ) ) x ^{6}}{( cos ( x ) e ^{x} ) ^{2}} : \frac{3 x ^{5} ( 6 cos ( x ) - x ( cos ( x ) - sin ( x ) ) )}{e ^{x} cos ^{2} ( x )}

= \frac{3 x ^{5} ( 6 cos ( x ) - x ( cos ( x ) - sin ( x ) ) )}{e ^{x} cos ^{2} ( x )}

a re a x^{2}, \frac{1}{2} x^{2}, 3 x

t an g e n t f (x) = x^{2} - 3, (- 2, 1)

\int \frac{1}{x ^{\frac{1}{2}} - 4} d x

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