derivative of (sin(x^2+1)^{1/y})

Lösung

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

e^{\frac{l n ( s i n ( x ^{2} ) + 1 )}{y}} (- \frac{ln ( sin ( x ^{2} ) + 1 ) \frac{d y}{d x}}{y ^{2}} + \frac{2 x cos ( x ^{2} )}{( sin ( x ^{2} ) + 1 ) y})

Schritte zur Lösung

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

Wende Exponentenregel an:

a^{b} = e^{b l n (a)}

(sin (x^{2}) + 1)^{\frac{1}{y}} = e^{\frac{1}{y} l n (s i n (x^{2}) + 1)}

= \frac{d}{d x} (e^{\frac{1}{y} l n (s i n (x^{2}) + 1)})

Wende die Kettenregel an:

e^{\frac{1}{y} l n (s i n (x^{2}) + 1)} \frac{d}{d x} (\frac{1}{y} ln (sin (x^{2}) + 1))

= e^{\frac{1}{y} l n (s i n (x^{2}) + 1)} \frac{d}{d x} (\frac{1}{y} ln (sin (x^{2}) + 1))

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

= e^{\frac{1}{y} l n (s i n (x^{2}) + 1)} (- \frac{ln ( sin ( x ^{2} ) + 1 ) \frac{d y}{d x}}{y ^{2}} + \frac{2 x cos ( x ^{2} )}{( sin ( x ^{2} ) + 1 ) y})

Multipliziere

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

= e^{\frac{l n ( s i n ( x ^{2} ) + 1 )}{y}} (- \frac{ln ( sin ( x ^{2} ) + 1 ) \frac{d y}{d x}}{y ^{2}} + \frac{2 x cos ( x ^{2} )}{( sin ( x ^{2} ) + 1 ) y})

\frac{\partial}{\partial x} (sin (π (6 x - 3 y)))

\frac{\partial}{\partial x} (x y - 5 x^{2} y)

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\int y e^{0.2 y} d y