Question

高数 高阶导数

这两个题怎么做 求解 谢谢

Answer 1

2. f(x) = ln(3+7x-6x^2), f' = (7-12x)/(3+7x-6x^2) = (7-12x)/[(3-2x)(1+3x)] = 3/(1+3x) - 2/(3-2x) = 3(1+3x)^(-1) - 2(3-2x)^(-1) f'' = 3*(-1)3(1+3x)^(-2) - 2(-1)(-2)(3-2x)^(-2), f''' = 3*(-1)(-2)3^2(1+3x)^(-3) - 2(-1)(-2)(-2)^2(3-2x)^(-3), ......................................................... f^(n) = 3(-1)^(n-1)(n-1)!3^(n-1)(1+3x)^(-n) - 2(-1)^(n-1)(n-1)!(-2)^(n-1)(3-2x)^(-n) = (-1)^(n-1) (n-1)! [3^n/(1+3x)^n + (-2)^n/(3-2x)^n] f^(n)(1) = (-1)^(n-1) (n-1)! [(3/4)^n + (-2)^n] 3. f(x) = (x-a)^n φ(x) 由莱布尼茨公式 f^(n)(x) = ∑C[(x-a)^n]^(n-k) φ^(k)(x), f^(n)(a) = ∑C[(a-a)^n]^(n-k) φ^(k)(a) 只有 k = n 时，[(a-a)^n]^(n-k) = 1， f^(n)(a) = φ^(n)(a)