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 :

$f(x,y)=g(y)+\tan^{-1}(xy)$

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

$\implies f_{xx}(x,y)=-\dfrac{2xy^3}{(1+(xy)^2)^2}$

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

$\implies f_{xx}(x,y)=\boxed{-\dfrac4{25}}$