Question

How do you use the definition of a derivative to find the derivative of `f(x)=5x`?

Answer 1

The limit definition of the derivative tells us that the derivative of a function is `f` is

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

For this, `f(x)=5x` and `f(x+h)=5(x+h)`, so:

`f'(x)=lim_(hrarr0)(5(x+h)-5x)/h`

`color(white)(f'(x))=lim_(hrarr0)(5x+5h-5x)/h`

`color(white)(f'(x))=lim_(hrarr0)(5h)/h`

`color(white)(f'(x))=lim_(hrarr0)5`

`color(white)(f'(x))=5`

So when `f(x)=5x`, we see that its derivative is `f'(x)=5`.