derivative of x^{3/2}log_{2}(sqrt(x+1))

Lösung

\frac{d}{d x} (x^{\frac{3}{2}} lo g_{2} (x + 1))

\frac{3 x ^{\frac{1}{2}} ln ( x + 1 ) ( x + 1 ) + 2 x ^{\frac{3}{2}}}{4 ln ( 2 ) ( x + 1 )}

Schritte zur Lösung

\frac{d}{d x} (x^{\frac{3}{2}} lo g_{2} (x + 1))

Wende die Produktregel an:

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

f = x^{\frac{3}{2}}, g = lo g_{2} (x + 1)

= \frac{d}{d x} (x^{\frac{3}{2}}) lo g_{2} (x + 1) + \frac{d}{d x} (lo g_{2} (x + 1)) x^{\frac{3}{2}}

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

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

= \frac{3 x ^{\frac{1}{2}}}{2} lo g_{2} (x + 1) + \frac{1}{2 ln ( 2 ) x + 1 ( x + 1 ) ^{\frac{1}{2}}} x^{\frac{3}{2}}

Vereinfache

\frac{3 x ^{\frac{1}{2}}}{2} lo g_{2} (x + 1) + \frac{1}{2 ln ( 2 ) x + 1 ( x + 1 ) ^{\frac{1}{2}}} x^{\frac{3}{2}} : \frac{3 x ^{\frac{1}{2}} ln ( x + 1 ) ( x + 1 ) + 2 x ^{\frac{3}{2}}}{4 ln ( 2 ) ( x + 1 )}

= \frac{3 x ^{\frac{1}{2}} ln ( x + 1 ) ( x + 1 ) + 2 x ^{\frac{3}{2}}}{4 ln ( 2 ) ( x + 1 )}

\int \frac{2}{3} x^{4} d x

\frac{\partial}{\partial x} (e^{y^{2}})

\int x (x + 1)^{100} d x

t an g e n t f (x) = e^{5 x}, at x = \frac{1}{5} ln (5)

x \to 4 - lim (\frac{1 - x}{4 - x})