derivative of 4^{9^{x^2}}

Lösung

\frac{d}{d x} (4^{9^{x^{2}}})

ln (2) ln (3) \cdot 9^{x^{2}} \cdot 2^{2 \cdot 9^{x^{2}} + 3} x

Schritte zur Lösung

\frac{d}{d x} (4^{9^{x^{2}}})

Wende Exponentenregel an:

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

4^{9^{x^{2}}} = e^{9^{x^{2}} l n (4)}

= \frac{d}{d x} (e^{9^{x^{2}} l n (4)})

Wende die Kettenregel an:

e^{9^{x^{2}} l n (4)} \frac{d}{d x} (9^{x^{2}} ln (4))

= e^{9^{x^{2}} l n (4)} \frac{d}{d x} (9^{x^{2}} ln (4))

\frac{d}{d x} (9^{x^{2}} ln (4)) = 8 ln (2) ln (3) \cdot 9^{x^{2}} x

= e^{9^{x^{2}} l n (4)} \cdot 8 ln (2) ln (3) \cdot 9^{x^{2}} x

Vereinfache

e^{9^{x^{2}} l n (4)} \cdot 8 ln (2) ln (3) \cdot 9^{x^{2}} x : ln (2) ln (3) \cdot 9^{x^{2}} \cdot 2^{2 \cdot 9^{x^{2}} + 3} x

= ln (2) ln (3) \cdot 9^{x^{2}} \cdot 2^{2 \cdot 9^{x^{2}} + 3} x

in v erse l a pl a ce \frac{1}{x} \cdot \frac{10}{x ^{2} + x + 10}

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

a re a y^{2} = 4 x, x = 1, x = 4

x \to 0 - lim (2 + \frac{3}{x})

\frac{d y}{d t} - 4 y = 10 sin (2 t), y (0) = 0