derivative of ln(x+sqrt(x^2+y^2))

Lösung

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

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

Schritte zur Lösung

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

\int \frac{x ^{6}}{x ^{7} - 2} d x

\int_{0}^{2} ((- x + 5) - (x^{2} - 9)) d x

\frac{d}{d x} (\frac{2 a}{x ^{3}})

\int \frac{12}{( x ^{2} - 9 )} d x

\frac{\partial}{\partial x} (- 6 x y - x^{6} - y^{6})