Question

Using the limit definition, how do you find the derivative of `f(x)=4x^2 -1`?

Answer 1

We get `f`(x)=8x` using our definition:

Explanation:

Let us start with our definition:
`f`(x)=lim_(h->0)(f(x+h)-f(x))/h`

where `h` is a small increment.
We can use our function into this definition:
`f`(x)=lim_(h->0)((4(x+h)^2-1)-(4x^2-1))/h=`

`=lim_(h->0)((4x^2+8xh+4h^2-1-4x^2+1)/h)=`

`=lim_(h->0)((8xh+4h^2)/h)=`

`=lim_(h->0)(4h(2x+h)/h)=`
cancel `h`
`=lim_(h->0)(4(2x+h))=`
as `h->0` we have:
`f`(x)=lim_(h->0)(4(2x+h))=8x`