Question
Obtain all the first and second order partial derivatives of the function:
f( x, y ) = In(I+xy^2)
f( x, y ) = In(I+xy^2)
Answers
Steve
∂f/∂x = 1/(1+xy^2) * y^2 = y^2/(1+xy^2)
∂f/∂y = 1/(1+xy^2) * 2xy = 2xy/(1+xy^2)
∂2f/∂x2 = -y^4/(1+xy^2)^2
∂2f/∂y^2 = 2x/(1+xy^2) - 4x^2y^2/(1+xy^2)^2
= 2x(1-xy^2)/(1+xy^2)^2
∂2f/∂x∂y = 2y/(1+xy^2) - y^2(2xy)/(1+xy^2)^2
= 2y/(1+xy^2)^2
∂2f/∂y∂x = 2y/(1+xy^2) - (2xy)(y^2)/(1+xy^2)^2
= 2y/(1+xy^2)^2
∂f/∂y = 1/(1+xy^2) * 2xy = 2xy/(1+xy^2)
∂2f/∂x2 = -y^4/(1+xy^2)^2
∂2f/∂y^2 = 2x/(1+xy^2) - 4x^2y^2/(1+xy^2)^2
= 2x(1-xy^2)/(1+xy^2)^2
∂2f/∂x∂y = 2y/(1+xy^2) - y^2(2xy)/(1+xy^2)^2
= 2y/(1+xy^2)^2
∂2f/∂y∂x = 2y/(1+xy^2) - (2xy)(y^2)/(1+xy^2)^2
= 2y/(1+xy^2)^2