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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

2 xx g (x) x^{x - 1} = 2 x^{x + 1} g (x)

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

d er i v a t i v e 3 x^{2} + 3 x

\frac{d}{d x} (2 ln (x) - \frac{x ^{2}}{16})

\int \frac{x}{9 x ^{2} - 4} d x

y^{^{''}} + 6 y^{^{'}} + 34 y = 68, y (0) = 5, y^{^{'}} (0) = 7

\int x^{0.813} d x