erweitern 5(3x^2-4x+1)^4 derivative of 3x^2-4x+1

Lösung

erweitern

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

405 x^{8} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 2160 x^{7} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 4860 x^{6} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 6000 x^{5} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 4430 x^{4} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 2000 x^{3} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 540 x^{2} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 80 x \frac{d}{d x} (3 x^{2} - 4 x + 1) + 5 \frac{d}{d x} (3 x^{2} - 4 x + 1)

Schritte zur Lösung

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

(3 x^{2} - 4 x + 1)^{4} = 81 x^{8} - 432 x^{7} + 972 x^{6} - 1200 x^{5} + 886 x^{4} - 400 x^{3} + 108 x^{2} - 16 x + 1

= 5 (81 x^{8} - 432 x^{7} + 972 x^{6} - 1200 x^{5} + 886 x^{4} - 400 x^{3} + 108 x^{2} - 16 x + 1) \frac{d}{d x} (3 x^{2} - 4 x + 1)

Setze Klammern

= 5 \frac{d}{d x} (3 x^{2} - 4 x + 1) 81 x^{8} + 5 \frac{d}{d x} (3 x^{2} - 4 x + 1) (- 432 x^{7}) + 5 \frac{d}{d x} (3 x^{2} - 4 x + 1) 972 x^{6} + 5 \frac{d}{d x} (3 x^{2} - 4 x + 1) (- 1200 x^{5}) + 5 \frac{d}{d x} (3 x^{2} - 4 x + 1) 886 x^{4} + 5 \frac{d}{d x} (3 x^{2} - 4 x + 1) (- 400 x^{3}) + 5 \frac{d}{d x} (3 x^{2} - 4 x + 1) 108 x^{2} + 5 \frac{d}{d x} (3 x^{2} - 4 x + 1) (- 16 x) + 5 \frac{d}{d x} (3 x^{2} - 4 x + 1) 1

Wende Minus-Plus Regeln an

+ (- a) = - a

= 5 \cdot 81 x^{8} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 5 \cdot 432 x^{7} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 5 \cdot 972 x^{6} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 5 \cdot 1200 x^{5} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 5 \cdot 886 x^{4} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 5 \cdot 400 x^{3} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 5 \cdot 108 x^{2} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 5 \cdot 16 x \frac{d}{d x} (3 x^{2} - 4 x + 1) + 5 \cdot 1 \frac{d}{d x} (3 x^{2} - 4 x + 1)

Vereinfache

5 \cdot 81 x^{8} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 5 \cdot 432 x^{7} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 5 \cdot 972 x^{6} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 5 \cdot 1200 x^{5} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 5 \cdot 886 x^{4} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 5 \cdot 400 x^{3} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 5 \cdot 108 x^{2} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 5 \cdot 16 x \frac{d}{d x} (3 x^{2} - 4 x + 1) + 5 \cdot 1 \frac{d}{d x} (3 x^{2} - 4 x + 1) : 405 x^{8} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 2160 x^{7} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 4860 x^{6} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 6000 x^{5} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 4430 x^{4} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 2000 x^{3} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 540 x^{2} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 80 x \frac{d}{d x} (3 x^{2} - 4 x + 1) + 5 \frac{d}{d x} (3 x^{2} - 4 x + 1)

= 405 x^{8} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 2160 x^{7} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 4860 x^{6} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 6000 x^{5} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 4430 x^{4} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 2000 x^{3} \frac{d}{d x} (3 x^{2} - 4 x + 1) + 540 x^{2} \frac{d}{d x} (3 x^{2} - 4 x + 1) - 80 x \frac{d}{d x} (3 x^{2} - 4 x + 1) + 5 \frac{d}{d x} (3 x^{2} - 4 x + 1)

e x p an d \frac{27 e ^{x} - 3 e ^{3 x}}{( 3 + e ^{x} ) ^{4}}

e x p an d (5 + 2 x^{\frac{1}{2}})^{4}

s im pl i f y (x + 2 - 3) \cdot (x + 2 + 3)

s im pl i f y 2 (2 + 3)

s im pl i f y 100.2 + \frac{28.9}{x ( 1 + x )}