Question

Y=X^2×e^(2X) 求Y的20阶导数

Answer 1

由莱布尼兹公式：
(uv)(n) = ∑[k=0,n] C[n,k] u(k) v(n-k)
[ x^2×e^(2x) ](n)
=∑[k=0,n] C[n,k] [x^2](k) [e^2x](n-k)
= C[n,0][x^2](0) [e^2x](n) + C[n,1][x^2](1) [e^2x](n-1)+ C[n,2][x^2](2) [e^2x](n-2)
+ ∑[k=n,3] C[n,k] [x^2](k) [e^2x](n-k)
= x^2*2^n*e^2x + n * 2x * 2^(n-1)*e^2x + n(n-1)/2 * 2 * 2^(n-2)e^2x
+ ∑[k=n,3] C[n,k]* 0* [e^2x](n-k)
= x^2*2^n*e^2x + n * 2x * 2^(n-1)*e^2x + n(n-1)/2 * 2 * 2^(n-2)e^2x