Question

What’s the difference quotient simplified

Answer 1

`\text{Use}\ (a+b)^3=a^3+3a^2b+3ab^2+b^3\\\\f(x+h)\to\text{exchange x to x + h}\\\\f(x)=x^3+3x\\\\f(x+h)=(x+h)^3+3(x+h)\\\\f(x+h)=x^3+3x^2h+3xh^2+h^3+3x+3h\\\\(f(x+h)-f(x))/(h)=(x^3+3x^2h+3xh^2+h^3+3x+3h-(x^3-3x))/(h)\\\\=(x^3+3x^2h+3xh^2+h^3+3x+3h-x^3-3x)/(h)\\\\=((x^3-x^3)+3x^2h+3xh^2+h^3+(3x-3x)+3h)/(h)\\\\=(3x^2h+3xh^2+h^3+3h)/(h)\\\\=(h(3x^2+3xh+h^2+3))/(h)\\\\=\boxed{3x^2+3xh+h^2+3}`