derivative (x^2-7x+8)sqrt(25-x^2)

Lösung

ableitung von

(x^{2} - 7 x + 8) 25 - x^{2}

\frac{- 3 x ^{3} + 14 x ^{2} + 42 x - 175}{25 - x ^{2}}

Schritte zur Lösung

\frac{d}{d x} ((x^{2} - 7 x + 8) 25 - x^{2})

Wende die Produktregel an:

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

f = x^{2} - 7 x + 8, g = 25 - x^{2}

= \frac{d}{d x} (x^{2} - 7 x + 8) 25 - x^{2} + \frac{d}{d x} (25 - x^{2}) (x^{2} - 7 x + 8)

\frac{d}{d x} (x^{2} - 7 x + 8) = 2 x - 7

\frac{d}{d x} (25 - x^{2}) = - \frac{x}{25 - x ^{2}}

= (2 x - 7) 25 - x^{2} + (- \frac{x}{25 - x ^{2}}) (x^{2} - 7 x + 8)

Vereinfache

(2 x - 7) 25 - x^{2} + (- \frac{x}{25 - x ^{2}}) (x^{2} - 7 x + 8) : \frac{- 3 x ^{3} + 14 x ^{2} + 42 x - 175}{25 - x ^{2}}

= \frac{- 3 x ^{3} + 14 x ^{2} + 42 x - 175}{25 - x ^{2}}

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