z=∫(xy1)e^t^2dt求他们一阶和二阶偏导数.一部分答案:δz/δx=ye^(x^2y^2)

Question

z=∫(xy 1)e^t^2dt
求他们一阶和二阶偏导数.一部分答案:δz/δx=ye^(x^2y^2)

Answer 1

对积分上限函数求导,就把积分上限代入函数中,再对上限求导,
所以
δz/δx
=e^(x²y²) *δ(xy)/δx
=y *e^(x²y²)
同理
δz/δy=x *e^(x²y²)
而二阶偏导数
δ²z/δx²
=δ[y *e^(x²y²)]/δx
=y *e^(x²y²) *δ(x²y²)/δx
=y *e^(x²y²) * 2xy²
=2xy^3 *e^(x²y²)
δ²z/δxδy
=δ[y *e^(x²y²)]/δy
=e^(x²y²) + y* 2x²y *e^(x²y²)
=(2x²y²+1) *e^(x²y²)
δ²z/δy²
=δ[x *e^(x²y²)] /δy
=x *e^(x²y²) *δ(x²y²)/δy
=x *e^(x²y²) *2x²y
=2x^3y *e^(x²y²)