derivative of x^a(1-x^b)

Lösung

\frac{d}{d x} (x^{a} (1 - x)^{b})

a x^{a - 1} (1 - x)^{b} - b x^{a} (1 - x)^{b - 1}

Schritte zur Lösung

\frac{d}{d x} (x^{a} (1 - x)^{b})

Wende die Produktregel an:

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

f = x^{a}, g = (1 - x)^{b}

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

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

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

= a x^{a - 1} (1 - x)^{b} + (- b (1 - x)^{b - 1}) x^{a}

Vereinfache

= a x^{a - 1} (1 - x)^{b} - b x^{a} (1 - x)^{b - 1}

x \to - 8 lim (\frac{- 5 x}{x + 8})

\int \frac{v}{0.122 - 0.0007 v ^{2}} d v

\int \frac{9 x ^{2} - 49}{x ^{3}} d x

l a pl a ce t r an s f or m 7 (t - 7)

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