derivative of 5^{(6x^2-7x(8x^2+9)})

Lösung

\frac{d}{d x} (5^{(6 x^{2} - 7 x) (8 x^{2} + 9)})

ln (5) \cdot 5^{(6 x^{2} - 7 x) (8 x^{2} + 9)} (192 x^{3} - 168 x^{2} + 108 x - 63)

Schritte zur Lösung

\frac{d}{d x} (5^{(6 x^{2} - 7 x) (8 x^{2} + 9)})

Wende Exponentenregel an:

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

5^{(6 x^{2} - 7 x) (8 x^{2} + 9)} = e^{(6 x^{2} - 7 x) (8 x^{2} + 9) l n (5)}

= \frac{d}{d x} (e^{(6 x^{2} - 7 x) (8 x^{2} + 9) l n (5)})

Wende die Kettenregel an:

e^{(6 x^{2} - 7 x) (8 x^{2} + 9) l n (5)} \frac{d}{d x} ((6 x^{2} - 7 x) (8 x^{2} + 9) ln (5))

= e^{(6 x^{2} - 7 x) (8 x^{2} + 9) l n (5)} \frac{d}{d x} ((6 x^{2} - 7 x) (8 x^{2} + 9) ln (5))

\frac{d}{d x} ((6 x^{2} - 7 x) (8 x^{2} + 9) ln (5)) = ln (5) (192 x^{3} - 168 x^{2} + 108 x - 63)

= e^{(6 x^{2} - 7 x) (8 x^{2} + 9) l n (5)} ln (5) (192 x^{3} - 168 x^{2} + 108 x - 63)

e^{(6 x^{2} - 7 x) (8 x^{2} + 9) l n (5)} = 5^{(6 x^{2} - 7 x) (8 x^{2} + 9)}

= ln (5) \cdot 5^{(6 x^{2} - 7 x) (8 x^{2} + 9)} (192 x^{3} - 168 x^{2} + 108 x - 63)