Choose a and take the derivative of g(x) at x= 1. The tangent equation is y=2x+ 1, and the tangent slope is g'(x).

That is g'(x)=2.

So f' (x) = g' (x)+x 2 = 2+x 2.

The slope at (1, f( 1)) is replaced by 1, and f'( 1)=4 is obtained.

You confuse the tangent equation with the tangent slope ... the derivative value of the function derivative is the tangent slope, not the diagonal itself.