derivative f(x)=(x^5+x^9)/x

Lösung

ableitung von

f (x) = \frac{x ^{5} + x ^{9}}{x}

8 x^{7} + 4 x^{3}

Schritte zur Lösung

\frac{d}{d x} (\frac{x ^{5} + x ^{9}}{x})

Vereinfache

\frac{x ^{5} + x ^{9}}{x} : x^{4} (x^{4} + 1)

= \frac{d}{d x} (x^{4} (x^{4} + 1))

Wende die Produktregel an:

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

f = x^{4}, g = x^{4} + 1

= \frac{d}{d x} (x^{4}) (x^{4} + 1) + \frac{d}{d x} (x^{4} + 1) x^{4}

\frac{d}{d x} (x^{4}) = 4 x^{3}

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

= 4 x^{3} (x^{4} + 1) + 4 x^{3} x^{4}

Vereinfache

4 x^{3} (x^{4} + 1) + 4 x^{3} x^{4} : 8 x^{7} + 4 x^{3}

= 8 x^{7} + 4 x^{3}

d er i v a t i v e y = 8 e^{- x} + e^{3 x}

x \to 0 lim (1 - \frac{cos ( x )}{x})

d er i v a t i v e \frac{5 + x - 5}{x}

d y - (y - 8)^{2} d x = 0

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