derivative of (3x^2/(x^3+1))

解答

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

\frac{3 ( - x ^{4} + 2 x )}{( x ^{3} + 1 ) ^{2}}

求解步骤

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

将常数提出:

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

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

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 ^{2} ) ( x ^{3} + 1 ) - \frac{d}{d x} ( x ^{3} + 1 ) x ^{2}}{( x ^{3} + 1 ) ^{2}}

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

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

= 3 \cdot \frac{2 x ( x ^{3} + 1 ) - 3 x ^{2} x ^{2}}{( x ^{3} + 1 ) ^{2}}

化简

3 \cdot \frac{2 x ( x ^{3} + 1 ) - 3 x ^{2} x ^{2}}{( x ^{3} + 1 ) ^{2}} : \frac{3 ( - x ^{4} + 2 x )}{( x ^{3} + 1 ) ^{2}}

= \frac{3 ( - x ^{4} + 2 x )}{( x ^{3} + 1 ) ^{2}}

\int \frac{3 ( 2 + ln ( - 3 x ) ) ^{3}}{x} d x

\int \frac{x ^{4} - 3 x + 3 x}{5 x ^{3}} d x

\int 18 d x

x \to 0 lim (\frac{x ^{5} + 4 x}{x})

d er i v a t i v e f (x) = \frac{1}{3} (x^{2} + 2)^{\frac{3}{2}}