derivative y=5y^2(4y^2+7)^{6/5}

解答

导数

y = 5 y^{2} (4 y^{2} + 7)^{\frac{6}{5}}

10 y (4 y^{2} + 7)^{\frac{6}{5}} + 48 y^{3} (4 y^{2} + 7)^{\frac{1}{5}}

求解步骤

\frac{d}{d y} (5 y^{2} (4 y^{2} + 7)^{\frac{6}{5}})

将常数提出:

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

= 5 \frac{d}{d y} (y^{2} (4 y^{2} + 7)^{\frac{6}{5}})

使用微分乘法定则:

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

f = y^{2}, g = (4 y^{2} + 7)^{\frac{6}{5}}

= 5 (\frac{d}{d y} (y^{2}) (4 y^{2} + 7)^{\frac{6}{5}} + \frac{d}{d y} ((4 y^{2} + 7)^{\frac{6}{5}}) y^{2})

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

\frac{d}{d y} ((4 y^{2} + 7)^{\frac{6}{5}}) = \frac{48 y ( 4 y ^{2} + 7 ) ^{\frac{1}{5}}}{5}

= 5 2 y (4 y^{2} + 7)^{\frac{6}{5}} + \frac{48 y ( 4 y ^{2} + 7 ) ^{\frac{1}{5}}}{5} y^{2}

化简

5 2 y (4 y^{2} + 7)^{\frac{6}{5}} + \frac{48 y ( 4 y ^{2} + 7 ) ^{\frac{1}{5}}}{5} y^{2} : 10 y (4 y^{2} + 7)^{\frac{6}{5}} + 48 y^{3} (4 y^{2} + 7)^{\frac{1}{5}}

= 10 y (4 y^{2} + 7)^{\frac{6}{5}} + 48 y^{3} (4 y^{2} + 7)^{\frac{1}{5}}

\frac{d y}{d x} = \frac{7 x}{3 y}

t an g e n t 4 x^{2} + 2 x

t an g e n t y = 5 x - x^{2}

\frac{d}{d x} (\frac{ln ( 4 x )}{x})

\frac{d}{d x} (x e^{1})