作业帮求反三角函数的导数y=rsinφ,φ=arcsin(y/r),求 dφ/dy.
来源：学生作业帮助网 编辑：作业帮 时间：2024/11/15 07:59:27
求反三角函数的导数y=rsinφ,φ=arcsin(y/r),求 dφ/dy.
求反三角函数的导数
y=rsinφ,
φ=arcsin(y/r),
求 dφ/dy.
求反三角函数的导数y=rsinφ,φ=arcsin(y/r),求 dφ/dy.
y = r sinφ
dy/dφ = r cosφ
因为 sinφ = y/r,所以 cosφ = (1-(y/r)^2)^(1/2)
dφ/dy = 1 / (dy/dφ) = 1 / ( r cosφ)
= 1 / [ r*(1-(y/r)^2)^(1/2) ]
dφ/dy=1/(dy/dφ)=1/(rcosφ)=1/[r(1-y^2/r^2)]^(1/2)
y=rsinφ,
φ=arcsin(y/r)
dφ/dr=1/√[1-(y/r)^2] *d(y/r)/dr
=1/√[1-(y/r)^2] *[(1/r)dy/dr+y(-1/r^2)/dr]
dφ/dy=1/√[1-(y/r)^2]*[(r-y)/r^2]
r=y/sinφ y/r=sinφ r-y=y(1/sinφ-1)=y(1...
全部展开
y=rsinφ,
φ=arcsin(y/r)
dφ/dr=1/√[1-(y/r)^2] *d(y/r)/dr
=1/√[1-(y/r)^2] *[(1/r)dy/dr+y(-1/r^2)/dr]
dφ/dy=1/√[1-(y/r)^2]*[(r-y)/r^2]
r=y/sinφ y/r=sinφ r-y=y(1/sinφ-1)=y(1-sinφ)/sinφ
r^2=y^2/(sinφ)^2
dφ/dy=(1/cosφ)*(y^2/(sinφ)^2)*y(1-sinφ)/sinφ
=2y^3(1-sinφ)/[sin2φ(sinφ)^2]
收起