Question

Find the difference quotient

Answer 1

Answer:

$\large\boxed{\dfrac{f(x+h)-f(x)}{h}=4x+2h}$

Step-by-step explanation:

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

$\dfrac{f(x+h)-f(x)}{h}=4x+2h$