(\partial)/(\partial x)(xcos(1/(x^2+y^2)))

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Wende die Produktregel an:

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

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

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

\frac{\partial}{\partial x} (x) = 1

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

= 1 \cdot cos (\frac{1}{x ^{2} + y ^{2}}) + \frac{2 x sin ( \frac{1}{x ^{2} + y ^{2}} )}{( x ^{2} + y ^{2} ) ^{2}} x

Vereinfache

1 \cdot cos (\frac{1}{x ^{2} + y ^{2}}) + \frac{2 x sin ( \frac{1}{x ^{2} + y ^{2}} )}{( x ^{2} + y ^{2} ) ^{2}} x : cos (\frac{1}{x ^{2} + y ^{2}}) + \frac{2 x ^{2} sin ( \frac{1}{x ^{2} + y ^{2}} )}{( x ^{2} + y ^{2} ) ^{2}}

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

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

\frac{\partial}{\partial x} (x + y cos (y))

d er i v a t i v e \frac{4 x ^{2} + 2 x + 2}{x}

\frac{d}{d x} (x^{2} - 7 x + 2)

\frac{d}{d x} (3 e^{x} + 5 x^{2})