If g(x, y) = x2 + y2 - 6x, find the gradient vector ∇g(1, 5)...?

Well,

edit : ... just to state something clearly...
all in all, I just guarantee the method in exact ... not the final result !!

here we have a typical example of what often happens
(either because I am at the office and do several things at a time ... and YAH maths does not have highest priority... or because it's close to midnight ... )

I rectified the final answer :
∇g(1,5) = <2*1 - 6, 2*5 > = < -4 , 10 > INSTEAD of the previous < -1, 10>
(thanks to great cidyah's help)
... just because 2*1 - 6 makes -4 and not -1 : that was the only error
So, let me just tell you that, in France, during an exam, this would count as a very minor error because the reasoning, the method, the way to apply it, etc. were all correct
but it seems that things look different in the USA, well at least nowadays, it seems, when there are less and less HUMAN TEACHERS ... !! ;-)

nevertheless, please don't take it wrong, but you should READ (go through) ALL MY ANSWER
I mean from the start till the end :
because if you cannot detect a simple arithmetic last line error as mine you have quite some work ahead, to say the least..

Mathematics is conducting a reasoning, under concepts, etc. ... and not just trying to feed a machine with correct answers.

Tangent's equation of g at point A(1,5) :
∇g(1,5) = < -1 , 10 >
slope at A is : m = 10/(-1) = -10
therefore, the tangent's equation is :
y - 5 = (-10)(x + 1)

y = -10x - 5 <--- tangent's equation

_____________________________________________________

g(x, y) = x2 + y2 - 6x
therefore :
∂g/∂x = 2x - 6
∂g/∂y = 2y
and so :
∇g(x,y) = < ∂g/∂x, ∂g/∂y > = <2x - 6 , 2y>
leading to :

∇g(1,5) = <2*1 - 6, 2*5 > = < -1 , 10 > <--- I added an intermediate step here !

hope it' ll help !!

g(x,y) = x^2+y^2 - 6x
∇g = <2x-6,2y>
∇g (1,5) = <-4 10>