derivative f(x)=(x+x^3)/(sqrt(x))

Lösung

ableitung von

f (x) = \frac{x + x ^{3}}{x}

\frac{1 + 5 x ^{2}}{2 x}

Schritte zur Lösung

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

Vereinfache

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

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

Wende die Produktregel an:

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

f = x, g = 1 + x^{2}

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

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

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

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

Vereinfache

\frac{1}{2 x} (1 + x^{2}) + 2 x x : \frac{1 + 5 x ^{2}}{2 x}

= \frac{1 + 5 x ^{2}}{2 x}

x \to 4 lim (\frac{5}{x - 3})

d er i v a t i v e y = \frac{x - 2}{x - 2}

\frac{\partial}{\partial x} (2 e^{5 x y})

\frac{\partial}{\partial x} (\frac{y ^{2}}{1 + x ^{2}})

(\frac{3 x ^{2}}{2})^{^{'}}