Question

How do you use the limit definition of the derivative to find the derivative of `f(x)=x^3+2x`?

Answer 1

`d/dx (x^3+2x) = 3x^2+2`

Explanation:

By definition:

`(df)/dx = lim_(h->0) (f(x+h)-f(x))/h`

so:

`d/dx (x^3+2x) = lim_(h->0) ((x+h)^3+2(x+h)-x^3-2x)/h`

Expand the cube of the binomial and simplify:

`d/dx (x^3+2x) = lim_(h->0) (color(blue)(x^3)+3x^2h+3xh^2+h^3 +color(red)(2x)+2h-color(blue)(x^3)-color(red)(2x))/h`

`d/dx (x^3+2x) = lim_(h->0) (3x^2h+3xh^2+h^3 +2h)/h`

`d/dx (x^3+2x) = lim_(h->0) (cancel(h)(3x^2+3xh+h^2 +2))/cancel(h)`

`d/dx (x^3+2x) = lim_(h->0) 3x^2+3xh+h^2 +2`

`d/dx (x^3+2x) = 3x^2+2`