derivative of x^2sin(1/(ax))

Lösung

\frac{d}{d x} (x^{2} sin (\frac{1}{a x}))

2 x sin (\frac{1}{a x}) - \frac{cos ( \frac{1}{a x} )}{a}

Schritte zur Lösung

\frac{d}{d x} (x^{2} sin (\frac{1}{a x}))

Wende die Produktregel an:

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

f = x^{2}, g = sin (\frac{1}{a x})

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

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

\frac{d}{d x} (sin (\frac{1}{a x})) = - \frac{cos ( \frac{1}{a x} )}{a x ^{2}}

= 2 x sin (\frac{1}{a x}) + (- \frac{cos ( \frac{1}{a x} )}{a x ^{2}}) x^{2}

Vereinfache

2 x sin (\frac{1}{a x}) + (- \frac{cos ( \frac{1}{a x} )}{a x ^{2}}) x^{2} : 2 x sin (\frac{1}{a x}) - \frac{cos ( \frac{1}{a x} )}{a}

= 2 x sin (\frac{1}{a x}) - \frac{cos ( \frac{1}{a x} )}{a}

x \to 1 - lim (\frac{2}{x ^{2} - 1})

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

x \to 4 lim (x^{2} + 9)

d er i v a t i v e f (x) = (5 + x^{4})^{3 x}

\frac{\partial}{\partial y} (x^{3} + k y^{2} - 5 x y)