derivative of arcsin(3xln(sqrt(x+1)))

Lösung

\frac{d}{d x} (arcsin (3 x) ln (x + 1))

\frac{3 ln ( x + 1 )}{2 1 - 9 x ^{2}} + \frac{arcsin ( 3 x )}{2 ( x + 1 )}

Schritte zur Lösung

\frac{d}{d x} (arcsin (3 x) ln (x + 1))

Wende die Produktregel an:

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

f = arcsin (3 x), g = ln (x + 1)

= \frac{d}{d x} (arcsin (3 x)) ln (x + 1) + \frac{d}{d x} (ln (x + 1)) arcsin (3 x)

\frac{d}{d x} (arcsin (3 x)) = \frac{3}{1 - 9 x ^{2}}

\frac{d}{d x} (ln (x + 1)) = \frac{1}{2 ( x + 1 )}

= \frac{3}{1 - 9 x ^{2}} ln (x + 1) + \frac{1}{2 ( x + 1 )} arcsin (3 x)

Vereinfache

\frac{3}{1 - 9 x ^{2}} ln (x + 1) + \frac{1}{2 ( x + 1 )} arcsin (3 x) : \frac{3 ln ( x + 1 )}{2 1 - 9 x ^{2}} + \frac{arcsin ( 3 x )}{2 ( x + 1 )}

= \frac{3 ln ( x + 1 )}{2 1 - 9 x ^{2}} + \frac{arcsin ( 3 x )}{2 ( x + 1 )}

\int 3 x zy + x y^{2} + 2 z x^{2}

\frac{d}{d x} (lo g_{x} (x^{2} + 1))

\int \frac{( x ^{2} + 4 )}{( x - 5 ) ( x ^{2} + 9 )} d x

\int x^{4} e^{(1 - x^{5})} d x

\frac{d y}{d x} + 5 y = 8