derivative of (4b/a xsqrt(a^2-x^2))

Lösung

\frac{d}{d x} (\frac{4 b}{a} x a^{2} - x^{2})

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

Schritte zur Lösung

\frac{d}{d x} (\frac{4 b}{a} x a^{2} - x^{2})

Entferne die Konstante:

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

= \frac{4 b}{a} \frac{d}{d x} (x a^{2} - x^{2})

Wende die Produktregel an:

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

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

= \frac{4 b}{a} (\frac{d x}{d x} a^{2} - x^{2} + \frac{d}{d x} (a^{2} - x^{2}) x)

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

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

= \frac{4 b}{a} (1 \cdot a^{2} - x^{2} + (- \frac{x}{a ^{2} - x ^{2}}) x)

Vereinfache

\frac{4 b}{a} (1 \cdot a^{2} - x^{2} + (- \frac{x}{a ^{2} - x ^{2}}) x) : \frac{4 b ( - 2 x ^{2} + a ^{2} )}{a a ^{2} - x ^{2}}

= \frac{4 b ( - 2 x ^{2} + a ^{2} )}{a a ^{2} - x ^{2}}

j o in 1 - 4 x + f (x)

d er i v a t i v e y = (arctan (x))^{2}

\frac{d}{d x} (\frac{e ^{x}}{x + 2})

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\int (8 x + 2)^{\frac{1}{3}} d x