derivative (x+4x)(x^{3/2}-x)

Lösung

ableitung von

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

\frac{5 ( 5 x ^{\frac{3}{2}} - 4 x )}{2}

Schritte zur Lösung

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

Vereinfache

(x + 4 x) (x^{\frac{3}{2}} - x) : 5 x (x^{\frac{3}{2}} - x)

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

Entferne die Konstante:

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

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

Wende die Produktregel an:

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

f = x, g = x^{\frac{3}{2}} - x

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

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

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

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

Vereinfache

5 (1 \cdot (x^{\frac{3}{2}} - x) + (\frac{3 x ^{\frac{1}{2}}}{2} - 1) x) : \frac{5 ( 5 x ^{\frac{3}{2}} - 4 x )}{2}

= \frac{5 ( 5 x ^{\frac{3}{2}} - 4 x )}{2}

\frac{\partial}{\partial x} (x e^{- z})

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