f'(x)=-6x and f'(-2)=12 the gradient or tangent slope.
If y=mx+c is the general equation of the tangent, m=gradient=12.
f(-2)=1-3*4=-11, so the tangent is at (-2,-11) which must lie on the tangent line, so:
-11=12(-2)+c, -11=-24+c, and c=13, so y=g(x)=12x+13 is the equation of the tangent.
DERIVATION OF DIFFERENTIAL
f(x+h)=1-3(x+h)^2=1-3x^2-6xh-3h^2.
∂f=f(x+h)-f(x)=-6xh-3h^2; ∂f/h=-6x-3h where ∂f is the change in f when x goes from x to x+h. h becomes ∂x, infinitesimally small, and can be ignored in comparison to x, so ∂f/∂x=f'(x)=-6x.