Find the difference quotient fx hfxh where h notequalto 0 f

Find the difference quotient f(x + h)-f(x)/h, where h notequalto 0, for the function below. f(x) = 4x^2 + 5x Simplify your answer as much as possible.

Solution

f(x) = 4x^2 + 5x

f(x+h) = 4 (x+h)^2 + 5 (x+h) = 4 (x^2 + 2hx+ h^2 ) + 5x + 5h

f(x+h) = 4x^2 + 8xh + 4h^2 + 5x + 5h

{ f(x+h) - f( x) } / h = 4x^2 + 8xh + 4h^2 + 5x + 5h - 4x^2 - 5x / h

= 8xh + 4h^2 + 5h / h

= 8x + 4h + 5