Question

How do you find the derivative using limits of `f(x)=-5x`?

Answer 1

Please see the explanation.

Explanation:

`f'(x) = lim_(hto0)(f(x +h) - f(x))/h`

given: `f(x) = -5x`

Then: `f(x + h) = -5(x + h) = -5x - 5h`

Substitute `-5x - 5h` for `f(x + h)`

`f'(x) = lim_(hto0)(-5x - 5h - f(x))/h`

Substitute `-5x` for `f(x)`

`f'(x) = lim_(hto0)(-5x - 5h - -5x)/h`

The x terms cancel

`f'(x) = lim_(hto0)(cancel(-5x) - 5h - cancel(-5x))/h`

The `h/h` cancels:

`f'(x) = lim_(hto0)(-5cancelh)/cancelh`

`f'(x) = lim_(hto0) -5`

Let the limit go to zero:

`f'(x) = -5`