Question

How do you use the definition of a derivative to find the derivative of `f ( x) = x^4`?

Answer 1

I found `4x^3`

Explanation:

The definition of derivative tells us that for a small increment `h` we can write:
`f'(x)=lim_(h->0)(f(x+h)-f(x))/h`
in our case:
`f'(x)=lim_(h->0)((x+h)^4-x^4)/h=`
`=lim_(h->0)(cancel(x^4)+6h^2x^2+4hx^3+4h^3x+h^4cancel(-x^4))/h=`
`=lim_(h->0)(cancel(h)(6hx^2+4x^3+4h^2x+h^3))/cancel(h)=`
as `h->0`

`=cancel(6hx^2)+4x^3+cancel(4h^2x)+cancel(h^3)=4x^3`