Question

If g (t) has a continuous second derivative and f(x,y)= g(y)+tan^-1(xy), then f_xx(2,1)=

Answer 1

Compute the partial derivatives with respect to x :

`f(x,y)=g(y)+\tan^(-1)(xy)`

`\implies f_x(x,y)=\frac y{1+(xy)^2}`

`\implies f_(xx)(x,y)=-(2xy^3)/((1+(xy)^2)^2)`

Evaluate the second partial derivative at (2, 1) :

`\implies f_(xx)(x,y)=\boxed{-\frac4{25}}`