derivative of arctan((x+y/(1-xy)))

Lösung

\frac{d}{d x} (arctan (\frac{x + y}{1 - x y}))

\frac{x ^{2} \frac{d y}{d x} + \frac{d y}{d x} + y ^{2} + 1}{x ^{2} + x ^{2} y ^{2} + y ^{2} + 1}

Schritte zur Lösung

\frac{d}{d x} (arctan (\frac{x + y}{1 - x y}))

Wende die Kettenregel an:

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

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

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

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

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

Vereinfache

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

= \frac{x ^{2} \frac{d y}{d x} + \frac{d y}{d x} + y ^{2} + 1}{x ^{2} + x ^{2} y ^{2} + y ^{2} + 1}

\frac{d}{d x} (4 x - 4)

\frac{\partial}{\partial y} (3 x^{2} (5 x + 7 y))

d er i v a t i v e f (x) = x^{3} - 6 x^{2} + 9 x

x \to \frac{π}{4} lim (tan (2 x))

\int \frac{7 x + 3}{x} d x