Question

How do you use the limit definition of the derivative to find the derivative of `f(x)=4x^2+1`?

Answer 1

`f'(x) =8x`

Explanation:

By definition `f'(x) =lim_(hrarr0)(f(x+h)-f(x))/h`

So, with `f(x)=4x^2+1` we have:

`f'(x) =lim_(hrarr0) ((4(x+h)^2+1 ) - (4x^2+1) ) / h`
`:. f'(x) =lim_(hrarr0) ((4(x^2+2hx+h^2)+1 ) - 4x^2-1 ) / h`
`:. f'(x) =lim_(hrarr0) (4x^2+8hx+4h^2+1 - 4x^2-1 ) / h`
`:. f'(x) =lim_(hrarr0) (8hx+4h^2) / h`
`:. f'(x) =lim_(hrarr0) (8x+4h)`
`:. f'(x) =8x`