y=Ⅹ x[(3x+1)^2(x-2)]^1/3的导数
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解决时间 2021-04-01 21:04
- 提问者网友：难遇难求
- 2021-04-01 01:38
y=Ⅹ x[(3x+1)^2(x-2)]^1/3的导数
最佳答案
- 五星知识达人网友：走死在岁月里
- 2021-04-01 03:04
y = x*(3x+1)² *(x-2)]^(1/3)
lny = lnx + 2ln(3x+1) + (1/3)ln(x-2)
1/y * y ′ = 1/x + 2/(3x+1) + 1/[3(x-2)]
y ′ = y * { 1/x + 2/(3x+1) + 1/[3(x-2)] }
= x*(3x+1)² *(x-2)]^(1/3) * { 1/x + 2/(3x+1) + 1/[3(x-2)] }
= (3x+1)² *(x-2)]^(1/3) + 2x*(3x+1) *(x-2)]^(1/3) + x*(3x+1)² /[3(x-2)]^(2/3) ]
= (3x+1){ (3x+1)(x-2) + 2x(x-2) + x(3x+1) } / [3(x-2)]^(2/3) ]
= (3x+1){ 3x²-5x-2 + 2x²-4x + 3x²+x } / [3(x-2)]^(2/3) ]
= (3x+1){ 8x²-8x-2} / [3(x-2)]^(2/3) ]
= 2(3x+1)(4x²-4x-1) / [3(x-2)]^(2/3) ]
lny = lnx + 2ln(3x+1) + (1/3)ln(x-2)
1/y * y ′ = 1/x + 2/(3x+1) + 1/[3(x-2)]
y ′ = y * { 1/x + 2/(3x+1) + 1/[3(x-2)] }
= x*(3x+1)² *(x-2)]^(1/3) * { 1/x + 2/(3x+1) + 1/[3(x-2)] }
= (3x+1)² *(x-2)]^(1/3) + 2x*(3x+1) *(x-2)]^(1/3) + x*(3x+1)² /[3(x-2)]^(2/3) ]
= (3x+1){ (3x+1)(x-2) + 2x(x-2) + x(3x+1) } / [3(x-2)]^(2/3) ]
= (3x+1){ 3x²-5x-2 + 2x²-4x + 3x²+x } / [3(x-2)]^(2/3) ]
= (3x+1){ 8x²-8x-2} / [3(x-2)]^(2/3) ]
= 2(3x+1)(4x²-4x-1) / [3(x-2)]^(2/3) ]
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