derivative 3^{sqrt(x^2+1)}

Lösung

ableitung von

3^{x^{2} + 1}

\frac{ln ( 3 ) \cdot 3 ^{x^{2} + 1} x}{x ^{2} + 1}

Schritte zur Lösung

\frac{d}{d x} (3^{x^{2} + 1})

Wende Exponentenregel an:

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

3^{x^{2} + 1} = e^{x^{2} + 1 l n (3)}

= \frac{d}{d x} (e^{x^{2} + 1 l n (3)})

Wende die Kettenregel an:

e^{x^{2} + 1 l n (3)} \frac{d}{d x} (x^{2} + 1 ln (3))

= e^{x^{2} + 1 l n (3)} \frac{d}{d x} (x^{2} + 1 ln (3))

\frac{d}{d x} (x^{2} + 1 ln (3)) = \frac{ln ( 3 ) x}{x ^{2} + 1}

= e^{x^{2} + 1 l n (3)} \frac{ln ( 3 ) x}{x ^{2} + 1}

Vereinfache

e^{x^{2} + 1 l n (3)} \frac{ln ( 3 ) x}{x ^{2} + 1} : \frac{ln ( 3 ) \cdot 3 ^{x^{2} + 1} x}{x ^{2} + 1}

= \frac{ln ( 3 ) \cdot 3 ^{x^{2} + 1} x}{x ^{2} + 1}

\frac{\partial}{\partial x} (x^{a} b^{b})

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t \to 3 lim (\frac{\frac{1}{t} - \frac{1}{3}}{t - 3})

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\frac{d}{d x} (4 π x^{2})