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

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = (x^{2} + 1)^{5}, g = (1 - x^{2})^{5}

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

\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}

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

Vereinfache

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

d er i v a t i v e f (x) = x^{3} - 2 x^{2} + 5

d er i v a t i v e f (x) = 3 x^{3} + 6 x^{2} + 7 x

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

\int 2 x sin (x) d x

y^{^{'}} + 8 y = t + e^{- 7 t}