(\partial)/(\partial x)(arctan(2x+y))

Lösung

\frac{\partial}{\partial x} (arctan (2 x + y))

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

Schritte zur Lösung

\frac{\partial}{\partial x} (arctan (2 x + y))

Behandle

y

als Konstante

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

y (1 + x^{2}) y^{^{'}} - x (4 + y^{2}) = 0

y^{^{'}} = \frac{2 x y ^{3}}{1 - x ^{2} y ^{2}}

f^{^{'}} (x) = - 3

x \to 0 lim (7 - x^{2})

x \to \infty lim (- x^{3})