正割的二阶导数Y=secx,y=cscx的二阶导数分别是什么

Question

正割的二阶导数
Y=secx,y=cscx的二阶导数分别是什么

Answer 1

1)
y = secx = 1/cosx =1/u =u^(-1)
dy/du = -u^(-2) = -1/u^2 = -1/(cosx)^2
du/dx = (cosx)' = -sinx
dy/dx = (dy/du)·(du/dx) = sinx/(cosx)^2 = sinx/(1-(sinx)^2) = m/(1-m^2)
d(dy/dx)/dm = [(1-m^2) - (-2m)·m]/(1-m^2)^2 = (1+m^2)/(1-m^2)^2 = (1+(sinx)^2)/(cosx)^4
dm/dx = (sinx)' = cosx
d^y/dx^2 = d(dy/dx)/dx = (d(dy/dx)/dm)·(dm/dx) = (1+(sinx)^2)/(cosx)^3
2)
y = cscx = 1/sinx = u^(-1)
dy/du = -1/u^2 = -1/(sinx)^2
du/dx = (sinx)' = cosx
dy/dx = (dy/du)·(du/dx) = -cosx/(1- (cosx)^2) = -m/(1-m^2)
d(dy/dx)/dm = [-(1-m^2) - (-2m)·(-m)]/(1-m^2)^2 = -(1+m^2)/(1-m^2)^2 = -(1+(cosx)^2)/(sinx)^4
dm/dx = (cosx)' = -sinx
d^2y/dx^2 = (d(dy/dx)/dm)·(dm/dx) = (1+(cosx)^2)/(sinx)^3