The value of the derivative at [tex]x=5[/tex] is the slope of the tangent line at the point [tex](x,g(x))=(5,-4)[/tex].
So the tangent line has equation
[tex]y-g(5)=g'(5)(x-5)\implies y+4=6(x-5)\implies\boxed{y=6x-34}[/tex]
