derivative of 2x(1-x^2^{1/2})

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

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

Wende die Produktregel an:

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

f = x, g = (1 - x^{2})^{\frac{1}{2}}

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

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

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

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

Vereinfache

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

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

x \frac{d y}{d x} + 2 y = 4 e^{x}

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