derivative f(x)=xsqrt(1-x^2)

Lösung

ableitung von

f (x) = x 1 - x^{2}

\frac{- 2 x ^{2} + 1}{1 - x ^{2}}

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = x, g = 1 - x^{2}

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

\frac{d x}{d x} = 1

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

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

Vereinfache

1 \cdot 1 - x^{2} + (- \frac{x}{1 - x ^{2}}) x : \frac{- 2 x ^{2} + 1}{1 - x ^{2}}

= \frac{- 2 x ^{2} + 1}{1 - x ^{2}}

\int_{0}^{x} 3 e^{- t^{2}} d t

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\int e^{x} 9 - e^{x} d x

\frac{d y}{d x} = - \frac{2 x y + 1}{x ^{2} + 2 y}