Question 389719
{{{x^2 - y^2 = 5}}}


Differentiating with respect to x,


{{{2x - 2y(dy/dx) = 0}}} Isolate dy/dx


{{{dy/dx = -2x/-2y = x/y}}}


Take the second derivative {{{d^2y/dx^2}}} using the Quotient rule:


{{{d^2y/dx^2 = (y - x(dy/dx))/(y^2)}}}


Replace {{{dy/dx}}} with {{{x/y}}}


{{{d^2y/dx^2 = (y - x^2/y)/y^2 = 1/y - x^2/y^3}}} (splitting up the fraction and simplifying)


Replacing x = 3 and y = 2, {{{d^2y/dx^2}}} at the point (3,2) is {{{1/2 - 9/8 = -5/8}}}