derivative of xsin(y/x)

Lösung

\frac{d}{d x} (x sin (\frac{y}{x}))

sin (\frac{y}{x}) + \frac{cos ( \frac{y}{x} ) ( x \frac{d y}{d x} - y )}{x}

Schritte zur Lösung

\frac{d}{d x} (x sin (\frac{y}{x}))

Wende die Produktregel an:

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

f = x, g = sin (\frac{y}{x})

= \frac{d x}{d x} sin (\frac{y}{x}) + \frac{d}{d x} (sin (\frac{y}{x})) x

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

\frac{d}{d x} (sin (\frac{y}{x})) = \frac{cos ( \frac{y}{x} ) ( x \frac{d y}{d x} - y )}{x ^{2}}

= 1 \cdot sin (\frac{y}{x}) + \frac{cos ( \frac{y}{x} ) ( x \frac{d y}{d x} - y )}{x ^{2}} x

Vereinfache

1 \cdot sin (\frac{y}{x}) + \frac{cos ( \frac{y}{x} ) ( x \frac{d y}{d x} - y )}{x ^{2}} x : sin (\frac{y}{x}) + \frac{cos ( \frac{y}{x} ) ( x \frac{d y}{d x} - y )}{x}

= sin (\frac{y}{x}) + \frac{cos ( \frac{y}{x} ) ( x \frac{d y}{d x} - y )}{x}

\frac{\partial}{\partial x} (cos (x) cos (y) e^{z})

\int \frac{1}{( x - 7 ) ^{2}} d x

y^{^{''}} - 5 y^{^{'}} - 6 y = 12 e^{2 t}

x \to 0 lim (\frac{1}{( x - 1 ) ^{2}} + 2)

\frac{d y}{d x} + 10 y = 11