(\partial)/(\partial x)(x*arctan(y/x))

Lösung

\frac{\partial}{\partial x} (x \cdot arctan (\frac{y}{x}))

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Wende die Produktregel an:

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

f = x, g = arctan (\frac{y}{x})

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

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

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

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

Vereinfache

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

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

l a pl a ce t r an s f or m 3 cos (3 t)

\int \frac{1}{x ^{2} - 8 x + 1} d x

d er i v a t i v e y = ln (7 x^{2} + 3 y^{2})

\int \frac{( 3 \cdot ln ( x ) - 5 ) ^{4}}{x} d x

\int_{1}^{6} t^{4} ln (4 t) d t