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

Lösung

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = x^{\frac{- 1}{2}}, g = sin (x)

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

\frac{d}{d x} (x^{\frac{- 1}{2}}) = - \frac{1}{2 x ^{\frac{3}{2}}}

\frac{d}{d x} (sin (x)) = cos (x)

= (- \frac{1}{2 x ^{\frac{3}{2}}}) sin (x) + cos (x) x^{\frac{- 1}{2}}

Vereinfache

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

= - \frac{sin ( x )}{2 x ^{\frac{3}{2}}} + \frac{cos ( x )}{x ^{\frac{1}{2}}}

\int 3 (3 x + 2) d x

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

x \to 2 π - lim (x csc (x))

\frac{d}{d x} (5 x^{2} + 4)

x \to - 5 lim (∣ x + 5 ∣)