derivative ((5x^2+9)^5)/((x-3)^4)

解答

导数

\frac{( 5 x ^{2} + 9 ) ^{5}}{( x - 3 ) ^{4}}

\frac{6 ( 5 x ^{2} + 9 ) ^{4} ( 5 x ^{2} - 25 x - 6 )}{( x - 3 ) ^{5}}

求解步骤

\frac{d}{d x} (\frac{( 5 x ^{2} + 9 ) ^{5}}{( x - 3 ) ^{4}})

使用微分除法定则:

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

= \frac{\frac{d}{d x} ( ( 5 x ^{2} + 9 ) ^{5} ) ( x - 3 ) ^{4} - \frac{d}{d x} ( ( x - 3 ) ^{4} ) ( 5 x ^{2} + 9 ) ^{5}}{( ( x - 3 ) ^{4} ) ^{2}}

\frac{d}{d x} ((5 x^{2} + 9)^{5}) = 50 x (5 x^{2} + 9)^{4}

\frac{d}{d x} ((x - 3)^{4}) = 4 (x - 3)^{3}

= \frac{50 x ( 5 x ^{2} + 9 ) ^{4} ( x - 3 ) ^{4} - 4 ( x - 3 ) ^{3} ( 5 x ^{2} + 9 ) ^{5}}{( ( x - 3 ) ^{4} ) ^{2}}

化简

\frac{50 x ( 5 x ^{2} + 9 ) ^{4} ( x - 3 ) ^{4} - 4 ( x - 3 ) ^{3} ( 5 x ^{2} + 9 ) ^{5}}{( ( x - 3 ) ^{4} ) ^{2}} : \frac{6 ( 5 x ^{2} + 9 ) ^{4} ( 5 x ^{2} - 25 x - 6 )}{( x - 3 ) ^{5}}

= \frac{6 ( 5 x ^{2} + 9 ) ^{4} ( 5 x ^{2} - 25 x - 6 )}{( x - 3 ) ^{5}}

\int x y^{3} d x

\int x^{3} tan^{3} (x^{4}) d x

f (x) = 3 x

y^{^{'}} + \frac{1}{x + 8} y = x^{- 2}, y (1) = 5

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