Question

请问截图中二阶偏导数的题如何计算？

为什么先求偏x再偏y和先偏y再偏x结果不同？

Answer 1

两种做法的结果是一样的，显然先求偏y要简单的多。

Answer 2

注意 e 的指数有 x， 要先代换掉。
令 t = u/√x , 则 dt = du/√x
f(x,y) = (1/√x)∫<0, yx^(3/2)>e^(u^2)du
f'x = [(-1/2)/x^(3/2)]∫<0, yx^(3/2)>e^(u^2)du + (3y/2)e^(y^2x^3)
f'y = xe^(y^2x^3)
f''xy = (-1/2)e^(y^2x^3) + (3/2)e^(y^2x^3) + 3y^2x^3e^(y^2x^3)
= (1+3y^2x^3)e^(y^2x^3)
f''yx = xe^(y^2x^3) = e^(y^2x^3) + 3y^2x^3e^(y^2x^3) = (1+3y^2x^3)e^(y^2x^3)
f''xy = f''yx