Calculating 3-Variable Derivative: Solving for dz/dt with x^2 + 3xt + 2t^2 = 1

[Shredface]

Homework Statement

Find dz/dt where z = (x^2)(t^2) and x^2 + 3xt + 2t^2 = 1.

2. The attempt at a solution

I really have no idea how to go about this, I've tried rearranging the second expression in terms of x but it's no help.

 

[HallsofIvy]

Use the impllicit differentiation.

Differentiating the first equation, with respect to t,
$$dz/dt= 2tx^2+ 2t^2x dx/dt$$

Differentiating the second equation, with respect to t,
$$2x dx/dt+ 3t dx/dt+ 3x+ 4t= 0$$
so
$$(2x+ 3t)dx/dt= -(3x+ 4t)$$
and
$$dx/dt= -(3x+4t)/(2x+ 3t)$$
Replace dx/dt in the first equation by that.