derivative of x^{1/2}(3x^2-35x+90)

解

\frac{d}{d x} (x^{\frac{1}{2}} (3 x^{2} - 35 x + 90))

\frac{15 x ^{2} - 105 x + 90}{2 x ^{\frac{1}{2}}}

解答ステップ

\frac{d}{d x} (x^{\frac{1}{2}} (3 x^{2} - 35 x + 90))

積の法則を適用:

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

f = x^{\frac{1}{2}}, g = 3 x^{2} - 35 x + 90

= \frac{d}{d x} (x^{\frac{1}{2}}) (3 x^{2} - 35 x + 90) + \frac{d}{d x} (3 x^{2} - 35 x + 90) x^{\frac{1}{2}}

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

\frac{d}{d x} (3 x^{2} - 35 x + 90) = 6 x - 35

= \frac{1}{2 x ^{\frac{1}{2}}} (3 x^{2} - 35 x + 90) + (6 x - 35) x^{\frac{1}{2}}

簡素化

\frac{1}{2 x ^{\frac{1}{2}}} (3 x^{2} - 35 x + 90) + (6 x - 35) x^{\frac{1}{2}} : \frac{15 x ^{2} - 105 x + 90}{2 x ^{\frac{1}{2}}}

= \frac{15 x ^{2} - 105 x + 90}{2 x ^{\frac{1}{2}}}