y=e^(-x)^2乘以cos20x的二阶导数

Question

y=e^(-x)^2乘以cos20x的二阶导数

Answer 1

y = e^(-x²) * cos(20x)
y' = e^(-x²) * (-x²)' * cos(20x) + (-sin(20x)) * (20x)' * e^(-x²)
= e^(-x²) * (-2x) * cos(20x) - sin(20x) * 20 * e^(-x²)
= -2e^(-x²) * [10sin(20x) + xcos(20x)]
d/dx [10sin(20x) + xcos(20x)]
= [cos(20x) + x * (-sin(20x)) * 20] + [10 * cos(20x) * 20]
= [cos(20x) - 20x * sin(20x)] + [200 * cos(20x)]
= 201cos(20x) - 20xsin(20x)
y'' = {e^(-x²) * [10sin(20x) + xcos(20x)]}',先省略 -2
= {[10sin(20x) + xcos(20x)] * e^(-x²) * (-2x)} - {e^(-x²) * [201cos(20x) - 20xsin(20x)]}
= -2xe^(-x²) * [10sin(20x) + xcos(20x)] - e^(-x²) * [20xsin(20x) - 201cos(20x)]
= e^(-x²) * [(201-2x²)cos(20x) - 40xsin(20x)],乘以之前省略的 -2,最後答案是
= -2[(201-2x²)cos(20x) - 40xsin(20x)] * e^(-x²)