derivative of e^{x/2}cos(x)

Lösung

\frac{d}{d x} (e^{\frac{x}{2}} cos (x))

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

Schritte zur Lösung

\frac{d}{d x} (e^{\frac{x}{2}} cos (x))

Wende die Produktregel an:

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

f = e^{\frac{x}{2}}, g = cos (x)

= \frac{d}{d x} (e^{\frac{x}{2}}) cos (x) + \frac{d}{d x} (cos (x)) e^{\frac{x}{2}}

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

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

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

Vereinfache

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

\frac{d y}{d x} = 9 x^{2} y

\int \frac{1}{25 x ^{2} - 36} d x

d er i v a t i v e 3 ln (3 x)

\int \frac{3 x ^{2} - 21 x + 32}{x ^{3} - 8 x ^{2} + 16 x} d x

y^{^{''}} + 4 y^{^{'}} + 4 y = 0, y (0) = 1, y^{^{'}} (0) = - 1