derivative y=sqrt(1+(2x+1)^8)

Lösung

ableitung von

y = 1 + (2 x + 1)^{8}

\frac{8 ( 128 x ^{7} + 448 x ^{6} + 672 x ^{5} + 560 x ^{4} + 280 x ^{3} + 84 x ^{2} + 14 x + 1 )}{1 + ( 2 x + 1 ) ^{8}}

Schritte zur Lösung

\frac{d}{d x} (1 + (2 x + 1)^{8})

Wende die Kettenregel an:

\frac{1}{2 1 + ( 2 x + 1 ) ^{8}} \frac{d}{d x} (1 + (2 x + 1)^{8})

= \frac{1}{2 1 + ( 2 x + 1 ) ^{8}} \frac{d}{d x} (1 + (2 x + 1)^{8})

\frac{d}{d x} (1 + (2 x + 1)^{8}) = 2048 x^{7} + 7168 x^{6} + 10752 x^{5} + 8960 x^{4} + 4480 x^{3} + 1344 x^{2} + 224 x + 16

= \frac{1}{2 1 + ( 2 x + 1 ) ^{8}} (2048 x^{7} + 7168 x^{6} + 10752 x^{5} + 8960 x^{4} + 4480 x^{3} + 1344 x^{2} + 224 x + 16)

Vereinfache

\frac{1}{2 1 + ( 2 x + 1 ) ^{8}} (2048 x^{7} + 7168 x^{6} + 10752 x^{5} + 8960 x^{4} + 4480 x^{3} + 1344 x^{2} + 224 x + 16) : \frac{8 ( 128 x ^{7} + 448 x ^{6} + 672 x ^{5} + 560 x ^{4} + 280 x ^{3} + 84 x ^{2} + 14 x + 1 )}{1 + ( 2 x + 1 ) ^{8}}

= \frac{8 ( 128 x ^{7} + 448 x ^{6} + 672 x ^{5} + 560 x ^{4} + 280 x ^{3} + 84 x ^{2} + 14 x + 1 )}{1 + ( 2 x + 1 ) ^{8}}

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