derivative f(x)=\sqrt[5]{(9x^7+3x^4-2)^3}

Lösung

ableitung von

f (x) = 5 (9 x^{7} + 3 x^{4} - 2)^{3}

\frac{3 ( 9 x ^{7} + 3 x ^{4} - 2 ) ^{2} ( 63 x ^{6} + 12 x ^{3} )}{5 ( ( 9 x ^{7} + 3 x ^{4} - 2 ) ^{3} ) ^{\frac{4}{5}}}

Schritte zur Lösung

\frac{d}{d x} (5 (9 x^{7} + 3 x^{4} - 2)^{3})

Wende die Kettenregel an:

\frac{1}{5 ( ( 9 x ^{7} + 3 x ^{4} - 2 ) ^{3} ) ^{\frac{4}{5}}} \frac{d}{d x} ((9 x^{7} + 3 x^{4} - 2)^{3})

= \frac{1}{5 ( ( 9 x ^{7} + 3 x ^{4} - 2 ) ^{3} ) ^{\frac{4}{5}}} \frac{d}{d x} ((9 x^{7} + 3 x^{4} - 2)^{3})

\frac{d}{d x} ((9 x^{7} + 3 x^{4} - 2)^{3}) = 3 (9 x^{7} + 3 x^{4} - 2)^{2} (63 x^{6} + 12 x^{3})

= \frac{1}{5 ( ( 9 x ^{7} + 3 x ^{4} - 2 ) ^{3} ) ^{\frac{4}{5}}} \cdot 3 (9 x^{7} + 3 x^{4} - 2)^{2} (63 x^{6} + 12 x^{3})

Vereinfache

\frac{1}{5 ( ( 9 x ^{7} + 3 x ^{4} - 2 ) ^{3} ) ^{\frac{4}{5}}} \cdot 3 (9 x^{7} + 3 x^{4} - 2)^{2} (63 x^{6} + 12 x^{3}) : \frac{3 ( 9 x ^{7} + 3 x ^{4} - 2 ) ^{2} ( 63 x ^{6} + 12 x ^{3} )}{5 ( ( 9 x ^{7} + 3 x ^{4} - 2 ) ^{3} ) ^{\frac{4}{5}}}

= \frac{3 ( 9 x ^{7} + 3 x ^{4} - 2 ) ^{2} ( 63 x ^{6} + 12 x ^{3} )}{5 ( ( 9 x ^{7} + 3 x ^{4} - 2 ) ^{3} ) ^{\frac{4}{5}}}

x \to 0 + lim (x^{\frac{4}{x}})

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