Tangent line of integral

We know that if $F(x) = \int_2^x g(t)dt$, then the slope of the tangent line to $F$ at a point $x_0$ is $$F’(x_0) = g(x_0)$$ by fundamental theorem of calculus.

Now suppose that we want to find the slope of the tangent line at $x_0 =1$, ($1$ is outside the limits of integration), is correct to say that $$F’(1) = - g(1) $$ is the slope of the tangente line?

Thanks.

Edit: $g$ continous on all reals.