(\partial)/(\partial x)(1/x cos(y/x))

Lösung

\frac{\partial}{\partial x} (\frac{1}{x} cos (\frac{y}{x}))

- \frac{cos ( \frac{y}{x} )}{x ^{2}} + \frac{y sin ( \frac{y}{x} )}{x ^{3}}

Schritte zur Lösung

\frac{\partial}{\partial x} (\frac{1}{x} cos (\frac{y}{x}))

Behandle

y

als Konstante

Wende die Produktregel an:

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

f = \frac{1}{x}, g = cos (\frac{y}{x})

= \frac{\partial}{\partial x} (\frac{1}{x}) cos (\frac{y}{x}) + \frac{\partial}{\partial x} (cos (\frac{y}{x})) \frac{1}{x}

\frac{\partial}{\partial x} (\frac{1}{x}) = - \frac{1}{x ^{2}}

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

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

Vereinfache

(- \frac{1}{x ^{2}}) cos (\frac{y}{x}) + \frac{y sin ( \frac{y}{x} )}{x ^{2}} \cdot \frac{1}{x} : - \frac{cos ( \frac{y}{x} )}{x ^{2}} + \frac{y sin ( \frac{y}{x} )}{x ^{3}}

= - \frac{cos ( \frac{y}{x} )}{x ^{2}} + \frac{y sin ( \frac{y}{x} )}{x ^{3}}

y^{^{'}} = (\frac{2 y + 3}{4 x + 5})^{2}

\int \frac{1}{( 25 x ^{2} + 4 ) ^{2}} d x

d er i v a t i v e f (x) = (3 x^{3} + 9 x + 7) (x^{2} + 8)

so l v e f or x, f = f e^{\frac{1}{f}}

x \to - 2 - lim ([x])