How do I find the derivatives of f and g?

[ice_cream_cone]

given that near (1,1,1) the curve of intersection of the surfaces:
x^6 + y^4 + z^7 -3xyz = 0 and
xy + yz + zx - 3z^4 = 0
has the paramteric equations x = f(t), y = g(t), z = t with f and g differentiable, how do i find rhe derivatives of f and g?
and the tangent line to the curve of intersection at (1,1,1)?

 

[amcavoy]

given that near (1,1,1) the curve of intersection of the surfaces:
x^6 + y^4 + z^7 -3xyz = 0 and
xy + yz + zx - 3z^4 = 0
has the paramteric equations x = f(t), y = g(t), z = t with f and g differentiable, how do i find rhe derivatives of f and g?
and the tangent line to the curve of intersection at (1,1,1)?

$$\vec{\mathbf{r}}=f(t)\,\hat{\mathbf{i}}+g(t)\,\hat{\mathbf{j}}+t\,\hat{\mathbf{k}}$$

What would happen if you plugged these values of x, y, and z into the original two equations?