slope of tangent line

Question

Use the limit definition of the derivative to find the instantaneous rate of change of  at 

Answer 1

f(x+h) = 2(x+h)^2 + 5(x+h) + 3 =

2(x^2+ 2xh + h^2) + 5(x+h) + 3 =

2x^2 + 4xh + 2h^2 + 5x + 5h + 3 =

numerator of difference quotient: 4xh+2h^2 + 5h

dividing by h: 4x + 2h + 5

limit as h-->0 is 4x+5

which agrees with the power rule

so f'(4) = 25