展开 dx(4x^4-6x^3+5x^2+3x)^4

解答

展开

d x (4 x^{4} - 6 x^{3} + 5 x^{2} + 3 x)^{4}

256 x^{16} d x - 1536 x^{15} d x + 4736 x^{14} d x - 8448 x^{13} d x + 8880 x^{12} d x - 3456 x^{11} d x - 2968 x^{10} d x + 4488 x^{9} d x - 671 x^{8} d x - 1308 x^{7} d x + 702 x^{6} d x + 540 x^{5} d x + 81 x^{4} d x

求解步骤

d x (4 x^{4} - 6 x^{3} + 5 x^{2} + 3 x)^{4}

(4 x^{4} - 6 x^{3} + 5 x^{2} + 3 x)^{4} = 256 x^{16} - 1536 x^{15} + 4736 x^{14} - 8448 x^{13} + 8880 x^{12} - 3456 x^{11} - 2968 x^{10} + 4488 x^{9} - 671 x^{8} - 1308 x^{7} + 702 x^{6} + 540 x^{5} + 81 x^{4}

= d x (256 x^{16} - 1536 x^{15} + 4736 x^{14} - 8448 x^{13} + 8880 x^{12} - 3456 x^{11} - 2968 x^{10} + 4488 x^{9} - 671 x^{8} - 1308 x^{7} + 702 x^{6} + 540 x^{5} + 81 x^{4})

打开括号

= d x \cdot 256 x^{16} + d x (- 1536 x^{15}) + d x \cdot 4736 x^{14} + d x (- 8448 x^{13}) + d x \cdot 8880 x^{12} + d x (- 3456 x^{11}) + d x (- 2968 x^{10}) + d x \cdot 4488 x^{9} + d x (- 671 x^{8}) + d x (- 1308 x^{7}) + d x \cdot 702 x^{6} + d x \cdot 540 x^{5} + d x \cdot 81 x^{4}

使用加减运算法则

+ (- a) = - a

= 256 x^{16} d x - 1536 x^{15} d x + 4736 x^{14} d x - 8448 x^{13} d x + 8880 x^{12} d x - 3456 x^{11} d x - 2968 x^{10} d x + 4488 x^{9} d x - 671 x^{8} d x - 1308 x^{7} d x + 702 x^{6} d x + 540 x^{5} d x + 81 x^{4} d x

e x p an d cos^{2} (x) (tan (x) + 1)^{2}

e x p an d (e^{7 x} + e^{- 7 x})^{2}

e x p an d (1 + x)^{7}

e x p an d (1 + x)^{6}

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