derivative of a{r}(xcos((x+1)/3))

Lösung

\frac{d}{d x} (a r (x) cos (\frac{x + 1}{3}))

a (cos (\frac{x + 1}{3}) \frac{d}{d x} (r (x)) - \frac{sin ( \frac{x + 1}{3} ) r ( x )}{3})

Schritte zur Lösung

\frac{d}{d x} (a r (x) cos (\frac{x + 1}{3}))

Entferne die Konstante:

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

= a \frac{d}{d x} (r (x) cos (\frac{x + 1}{3}))

Wende die Produktregel an:

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

f = r (x), g = cos (\frac{x + 1}{3})

= a (\frac{d}{d x} (r (x)) cos (\frac{x + 1}{3}) + \frac{d}{d x} (cos (\frac{x + 1}{3})) r (x))

\frac{d}{d x} (cos (\frac{x + 1}{3})) = - sin (\frac{x + 1}{3}) \frac{1}{3}

= a (\frac{d}{d x} (r (x)) cos (\frac{x + 1}{3}) + (- sin (\frac{x + 1}{3}) \frac{1}{3}) r (x))

Vereinfache

a (\frac{d}{d x} (r (x)) cos (\frac{x + 1}{3}) + (- sin (\frac{x + 1}{3}) \frac{1}{3}) r (x)) : a (cos (\frac{x + 1}{3}) \frac{d}{d x} (r (x)) - \frac{sin ( \frac{x + 1}{3} ) r ( x )}{3})

= a (cos (\frac{x + 1}{3}) \frac{d}{d x} (r (x)) - \frac{sin ( \frac{x + 1}{3} ) r ( x )}{3})

\frac{d}{d x} (18 x - \frac{2}{3} x^{3})

x \to 1 lim (e^{x} - e)

\int_{3}^{\infty} x e^{- x} d x

\int \frac{1}{x ^{4} x ^{2} - 2} d x

\frac{d}{d x} (\frac{cos ( 3 x ^{3} )}{tan ( 3 x ^{2} + 2 x )})