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

Lösung

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

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

\int \frac{7 x ^{2} + 5 x - 6}{( x ^{2} - 4 ) ( x + 2 )} d x

f (x) = x^{4 x}

x \to 1 lim (- 6 x sin (3 x^{2} + 1))

d er i v a t i v e f (x) = \frac{4 x}{x + 2}

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