derivative (4x)^{10x^3}

Lösung

ableitung von

(4 x)^{10 x^{3}}

15 ln (2) \cdot 2^{20 x^{3} + 2} x^{2 + 10 x^{3}} + 5 \cdot 2^{1 + 20 x^{3}} x^{10 x^{3}} (3 x^{2} ln (x) + x^{2})

Schritte zur Lösung

\frac{d}{d x} ((4 x)^{10 x^{3}})

Vereinfache

(4 x)^{10 x^{3}} : 104857 6^{x^{3}} x^{10 x^{3}}

= \frac{d}{d x} (104857 6^{x^{3}} x^{10 x^{3}})

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = 104857 6^{x^{3}}, g = x^{10 x^{3}}

= \frac{d}{d x} (104857 6^{x^{3}}) x^{10 x^{3}} + \frac{d}{d x} (x^{10 x^{3}}) 104857 6^{x^{3}}

\frac{d}{d x} (104857 6^{x^{3}}) = 15 ln (2) \cdot 2^{20 x^{3} + 2} x^{2}

\frac{d}{d x} (x^{10 x^{3}}) = 10 x^{10 x^{3}} (3 x^{2} ln (x) + x^{2})

= 15 ln (2) \cdot 2^{20 x^{3} + 2} x^{2} x^{10 x^{3}} + 10 x^{10 x^{3}} (3 x^{2} ln (x) + x^{2}) \cdot 104857 6^{x^{3}}

Vereinfache

15 ln (2) \cdot 2^{20 x^{3} + 2} x^{2} x^{10 x^{3}} + 10 x^{10 x^{3}} (3 x^{2} ln (x) + x^{2}) \cdot 104857 6^{x^{3}} : 15 ln (2) \cdot 2^{20 x^{3} + 2} x^{2 + 10 x^{3}} + 5 \cdot 2^{1 + 20 x^{3}} x^{10 x^{3}} (3 x^{2} ln (x) + x^{2})

= 15 ln (2) \cdot 2^{20 x^{3} + 2} x^{2 + 10 x^{3}} + 5 \cdot 2^{1 + 20 x^{3}} x^{10 x^{3}} (3 x^{2} ln (x) + x^{2})

\frac{d}{d x} (\frac{1}{sinh ( x )})

\frac{d}{d x} (\frac{d ^{2}}{d x ^{2}} (\frac{1}{x}) - ln (x))

x \to 0 lim ((\frac{4}{x} - 1) x)

d er i v a t i v e 5 x + x^{2}

\frac{d y}{d t} = 3 t^{2}