derivative f(x)= 1/27 (9x^2+6)^{3/2}

解答

导数

f (x) = \frac{1}{27} (9 x^{2} + 6)^{\frac{3}{2}}

x (9 x^{2} + 6)^{\frac{1}{2}}

求解步骤

\frac{d}{d x} (\frac{1}{27} (9 x^{2} + 6)^{\frac{3}{2}})

将常数提出:

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

= \frac{1}{27} \frac{d}{d x} ((9 x^{2} + 6)^{\frac{3}{2}})

使用连锁律:

\frac{3 ( 9 x ^{2} + 6 ) ^{\frac{1}{2}}}{2} \frac{d}{d x} (9 x^{2} + 6)

= \frac{3 ( 9 x ^{2} + 6 ) ^{\frac{1}{2}}}{2} \frac{d}{d x} (9 x^{2} + 6)

\frac{d}{d x} (9 x^{2} + 6) = 18 x

= \frac{1}{27} \cdot \frac{3 ( 9 x ^{2} + 6 ) ^{\frac{1}{2}}}{2} \cdot 18 x

化简

\frac{1}{27} \cdot \frac{3 ( 9 x ^{2} + 6 ) ^{\frac{1}{2}}}{2} \cdot 18 x : x (9 x^{2} + 6)^{\frac{1}{2}}

= x (9 x^{2} + 6)^{\frac{1}{2}}