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

Lösung

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

\frac{d x}{d x} = 1

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

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

Vereinfache

1 \cdot (x^{2} + 3)^{\frac{1}{2}} + \frac{x}{( x ^{2} + 3 ) ^{\frac{1}{2}}} x : \frac{2 x ^{2} + 3}{( x ^{2} + 3 ) ^{\frac{1}{2}}}

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

\frac{\partial}{\partial y} (x^{3} - 3 x + 3 x y^{2})

d er i v a t i v e f (x) = 7 - x^{2}

3 x^{2} y d x + y d y = 0

x \to 1 lim (x + \frac{1}{x})

t an g e n t y = \frac{1}{x ^{3}}, (- 3, - \frac{1}{27})