derivative of ar(xcot(x/y))

Lösung

\frac{d}{d x} (a r (x) cot (\frac{x}{y}))

a cot (\frac{x}{y}) \frac{d}{d x} (r (x)) - \frac{csc ^{2} ( \frac{x}{y} ) ( y - x \frac{d y}{d x} ) r ( x )}{y ^{2}}

Schritte zur Lösung

\frac{d}{d x} (a r (x) cot (\frac{x}{y}))

Entferne die Konstante:

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

= a \frac{d}{d x} (r (x) cot (\frac{x}{y}))

Wende die Produktregel an:

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

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

= a (\frac{d}{d x} (r (x)) cot (\frac{x}{y}) + \frac{d}{d x} (cot (\frac{x}{y})) r (x))

\frac{d}{d x} (cot (\frac{x}{y})) = - \frac{csc ^{2} ( \frac{x}{y} ) ( y - x \frac{d y}{d x} )}{y ^{2}}

= a \frac{d}{d x} (r (x)) cot (\frac{x}{y}) + - \frac{csc ^{2} ( \frac{x}{y} ) ( y - x \frac{d y}{d x} )}{y ^{2}} r (x)

Vereinfache

a \frac{d}{d x} (r (x)) cot (\frac{x}{y}) + - \frac{csc ^{2} ( \frac{x}{y} ) ( y - x \frac{d y}{d x} )}{y ^{2}} r (x) : a cot (\frac{x}{y}) \frac{d}{d x} (r (x)) - \frac{csc ^{2} ( \frac{x}{y} ) ( y - x \frac{d y}{d x} ) r ( x )}{y ^{2}}

= a cot (\frac{x}{y}) \frac{d}{d x} (r (x)) - \frac{csc ^{2} ( \frac{x}{y} ) ( y - x \frac{d y}{d x} ) r ( x )}{y ^{2}}