derivative y=((x^2+1)^5)/((1-x^2)^5)

Решение

производное от

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

\frac{20 x ( x ^{2} + 1 ) ^{4}}{( 1 - x ^{2} ) ^{6}}

Шаги решения

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

Производная частного:

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

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

\frac{d}{d x} ((x^{2} + 1)^{5}) = 10 x (x^{2} + 1)^{4}

\frac{d}{d x} ((1 - x^{2})^{5}) = - 10 x (1 - x^{2})^{4}

= \frac{10 x ( x ^{2} + 1 ) ^{4} ( 1 - x ^{2} ) ^{5} - ( - 10 x ( 1 - x ^{2} ) ^{4} ) ( x ^{2} + 1 ) ^{5}}{( ( 1 - x ^{2} ) ^{5} ) ^{2}}

Упростите

\frac{10 x ( x ^{2} + 1 ) ^{4} ( 1 - x ^{2} ) ^{5} - ( - 10 x ( 1 - x ^{2} ) ^{4} ) ( x ^{2} + 1 ) ^{5}}{( ( 1 - x ^{2} ) ^{5} ) ^{2}} : \frac{20 x ( x ^{2} + 1 ) ^{4}}{( 1 - x ^{2} ) ^{6}}

= \frac{20 x ( x ^{2} + 1 ) ^{4}}{( 1 - x ^{2} ) ^{6}}