Question

How do you find f'(x) using the limit definition given `4x^2 -1`?

Answer 1

When a question asks you to differentiate using the definition of the derivative, it means to use the formula `f'(x) =lim_(h-> 0) (f(x + h) - f(x))/h`.

Explanation:

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

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

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

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

`f'(x) = lim_(h-> 0)(cancel(h)(8x + 4h))/cancel(h)`

`f'(x) = 8x + 4(0)`

`f'(x) = 8x`

The derivative is therefore `f'(x) = 8x`