(d^8)/(dx^8)(sqrt(1-x^2))

解答

\frac{d ^{8}}{d x ^{8}} (1 - x^{2})

- 45 \cdot \frac{( - 792 x ^{6} ( - x ^{2} + 1 ) ^{\frac{7}{2}} + 88 x ^{4} ( - x ^{2} + 1 ) ^{\frac{9}{2}} - 1188 x ^{4} ( - x ^{2} + 1 ) ^{\frac{7}{2}} + 96 x ^{2} ( - x ^{2} + 1 ) ^{\frac{11}{2}} + 132 x ^{2} ( - x ^{2} + 1 ) ^{\frac{9}{2}} - 99 x ^{2} ( - x ^{2} + 1 ) ^{\frac{7}{2}} + 24 ( - x ^{2} + 1 ) ^{\frac{11}{2}} + 11 ( - x ^{2} + 1 ) ^{\frac{9}{2}} ) ( ( 1 - x ^{2} ) ^{\frac{11}{2}} ) ^{2} - ( - 22 x ( 1 - x ^{2} ) ^{10} ) ( ( 32 x ^{3} + 24 x ) ( 1 - x ^{2} ) ^{\frac{11}{2}} - ( - 11 x ( 1 - x ^{2} ) ^{\frac{9}{2}} ) ( 8 x ^{4} + 12 x ^{2} + 1 ) )}{( ( ( 1 - x ^{2} ) ^{\frac{11}{2}} ) ^{2} ) ^{2}}

求解步骤

\frac{d ^{8}}{d x ^{8}} (1 - x^{2})

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

= \frac{d ^{7}}{d x ^{7}} (- \frac{x}{1 - x ^{2}})

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

= \frac{d ^{6}}{d x ^{6}} (- \frac{1}{( 1 - x ^{2} ) 1 - x ^{2}})

\frac{d}{d x} (- \frac{1}{( 1 - x ^{2} ) 1 - x ^{2}}) = - \frac{3 x}{( 1 - x ^{2} ) ^{\frac{5}{2}}}

= \frac{d ^{5}}{d x ^{5}} (- \frac{3 x}{( 1 - x ^{2} ) ^{\frac{5}{2}}})

\frac{d}{d x} (- \frac{3 x}{( 1 - x ^{2} ) ^{\frac{5}{2}}}) = - \frac{3 ( 4 x ^{2} + 1 )}{( 1 - x ^{2} ) ^{\frac{7}{2}}}

= \frac{d ^{4}}{d x ^{4}} (- \frac{3 ( 4 x ^{2} + 1 )}{( 1 - x ^{2} ) ^{\frac{7}{2}}})

\frac{d}{d x} (- \frac{3 ( 4 x ^{2} + 1 )}{( 1 - x ^{2} ) ^{\frac{7}{2}}}) = - \frac{15 x ( 4 x ^{2} + 3 )}{( 1 - x ^{2} ) ^{\frac{9}{2}}}

= \frac{d ^{3}}{d x ^{3}} (- \frac{15 x ( 4 x ^{2} + 3 )}{( 1 - x ^{2} ) ^{\frac{9}{2}}})

\frac{d}{d x} (- \frac{15 x ( 4 x ^{2} + 3 )}{( 1 - x ^{2} ) ^{\frac{9}{2}}}) = - \frac{45 ( 8 x ^{4} + 12 x ^{2} + 1 )}{( 1 - x ^{2} ) ^{\frac{11}{2}}}

= \frac{d ^{2}}{d x ^{2}} (- \frac{45 ( 8 x ^{4} + 12 x ^{2} + 1 )}{( 1 - x ^{2} ) ^{\frac{11}{2}}})

\frac{d}{d x} (- \frac{45 ( 8 x ^{4} + 12 x ^{2} + 1 )}{( 1 - x ^{2} ) ^{\frac{11}{2}}}) = - 45 \cdot \frac{( 32 x ^{3} + 24 x ) ( 1 - x ^{2} ) ^{\frac{11}{2}} - ( - 11 x ( 1 - x ^{2} ) ^{\frac{9}{2}} ) ( 8 x ^{4} + 12 x ^{2} + 1 )}{( ( 1 - x ^{2} ) ^{\frac{11}{2}} ) ^{2}}

= \frac{d}{d x} - 45 \cdot \frac{( 32 x ^{3} + 24 x ) ( 1 - x ^{2} ) ^{\frac{11}{2}} - ( - 11 x ( 1 - x ^{2} ) ^{\frac{9}{2}} ) ( 8 x ^{4} + 12 x ^{2} + 1 )}{( ( 1 - x ^{2} ) ^{\frac{11}{2}} ) ^{2}}

\frac{d}{d x} - 45 \cdot \frac{( 32 x ^{3} + 24 x ) ( 1 - x ^{2} ) ^{\frac{11}{2}} - ( - 11 x ( 1 - x ^{2} ) ^{\frac{9}{2}} ) ( 8 x ^{4} + 12 x ^{2} + 1 )}{( ( 1 - x ^{2} ) ^{\frac{11}{2}} ) ^{2}} = - 45 \cdot \frac{( - 792 x ^{6} ( - x ^{2} + 1 ) ^{\frac{7}{2}} + 88 x ^{4} ( - x ^{2} + 1 ) ^{\frac{9}{2}} - 1188 x ^{4} ( - x ^{2} + 1 ) ^{\frac{7}{2}} + 96 x ^{2} ( - x ^{2} + 1 ) ^{\frac{11}{2}} + 132 x ^{2} ( - x ^{2} + 1 ) ^{\frac{9}{2}} - 99 x ^{2} ( - x ^{2} + 1 ) ^{\frac{7}{2}} + 24 ( - x ^{2} + 1 ) ^{\frac{11}{2}} + 11 ( - x ^{2} + 1 ) ^{\frac{9}{2}} ) ( ( 1 - x ^{2} ) ^{\frac{11}{2}} ) ^{2} - ( - 22 x ( 1 - x ^{2} ) ^{10} ) ( ( 32 x ^{3} + 24 x ) ( 1 - x ^{2} ) ^{\frac{11}{2}} - ( - 11 x ( 1 - x ^{2} ) ^{\frac{9}{2}} ) ( 8 x ^{4} + 12 x ^{2} + 1 ) )}{( ( ( 1 - x ^{2} ) ^{\frac{11}{2}} ) ^{2} ) ^{2}}

= - 45 \cdot \frac{( - 792 x ^{6} ( - x ^{2} + 1 ) ^{\frac{7}{2}} + 88 x ^{4} ( - x ^{2} + 1 ) ^{\frac{9}{2}} - 1188 x ^{4} ( - x ^{2} + 1 ) ^{\frac{7}{2}} + 96 x ^{2} ( - x ^{2} + 1 ) ^{\frac{11}{2}} + 132 x ^{2} ( - x ^{2} + 1 ) ^{\frac{9}{2}} - 99 x ^{2} ( - x ^{2} + 1 ) ^{\frac{7}{2}} + 24 ( - x ^{2} + 1 ) ^{\frac{11}{2}} + 11 ( - x ^{2} + 1 ) ^{\frac{9}{2}} ) ( ( 1 - x ^{2} ) ^{\frac{11}{2}} ) ^{2} - ( - 22 x ( 1 - x ^{2} ) ^{10} ) ( ( 32 x ^{3} + 24 x ) ( 1 - x ^{2} ) ^{\frac{11}{2}} - ( - 11 x ( 1 - x ^{2} ) ^{\frac{9}{2}} ) ( 8 x ^{4} + 12 x ^{2} + 1 ) )}{( ( ( 1 - x ^{2} ) ^{\frac{11}{2}} ) ^{2} ) ^{2}}

in v erse l a pl a ce \frac{3 s}{( s ^{2} + 1 )}

e x t re m e f (x) = r x + n

\frac{\partial}{\partial t} (t - s)

d er i v a t i v e - 2 x^{4} + 5 x^{2} - 3

d er i v a t i v e f (x) = e^{4 t s i n (2 t)}