f'(x)=x*1/(2sqrt(x-1))+sqrt(x-1)
simplify
f'(x)=x/(2sqrt(x-1))+sqrt(x-1)
f'(x)=x/(2sqrt(x-1))+sqrt(x-1)*(2sqrt(x-1))/(2sqrt(x-1))
f'(x)=x/(2sqrt(x-1))+(2(x-1))/(2sqrt(x-1))
f'(x)=(x+2(x-1))/(2sqrt(x-1))
expand the brackets
f'(x)=(x+2x-2)/(2sqrt(x-1))
f'(x)=(3x-2)/(2sqrt(x-1))
plug into point-slope formula
y-y_1=f'(x_1)(x-x_1)
remembering that the derivative finds the slope of a function so m=f'(x_1)
to find y_1
f(4)=(4)sqrt((4)-1)
y_1=4sqrt(3)
to find f'(x_1)
f'(4)=(3(4)-2)/(2sqrt((4)-1)
f'(4)=(12-2)/(2sqrt(3))
f'(4)=10/(2sqrt(3))
f'(4)=5/sqrt(3)
plug in
y-4sqrt(3)=5/sqrt(3)(x-4)
add 4sqrt(3) to each side
ycancel(-4sqrt(3)+4sqrt(3))=5/sqrt(3)(x-4)+4sqrt3
expand brackets
y=5/sqrt(3)x-5/sqrt3(4)
y=5/sqrt3x-20/sqrt3