Question

Find the difference quotient

Answer 1

`\large\boxed{(f(x+h)-f(x))/(h)=4x+2h}`

Explanation:

`f(x)=2x^2+4\\\\f(x+h)=2(x+h)^2+4\qquad\text{use}\ (a+b)^2=a^2+2ab+b^2\\\\f(x+h)=2(x^2+2xh+h^2)+4\qquad\text{use the distributive property}\\\\f(x+h)=2x^2+4xh+2h^2+4\\\\\text{Substitute to}\ (f(x+h)-f(x))/(h)\\\\(f(x+h)-f(x))/(h)=(2x^2+4xh+2h^2+4-(2x^2+4))/(h)\\\\(f(x+h)-f(x))/(h)=(2x^2+4xh+2h^2+4-2x^2-4)/(h)\\\\\text{combine like terms}\\\\(f(x+h)-f(x))/(h)=(4xh+2h^2)/(h)\qquad\text{distribute}\\\\(f(x+h)-f(x))/(h)=(h(4x+2h))/(h)\qquad\text{cancel}\ h`

`(f(x+h)-f(x))/(h)=4x+2h`