Question

How do you use the limit definition of the derivative to find the derivative of `f(x)=5x+1`?

Answer 1

`f'(x) = 5`

Explanation:

Limit definition of derivative:
`f'(x) = limh->0 ((f(x+h)-f(x))/h)`

Plugging in our function `f(x) = 5x+1`
`f'(x) = limh->0 [((5(x+h)+1)-(5x+1))/h]`

Distributing terms in the numerator:
`f'(x) = limh->0 [(5x+5h+1-5x-1)/h]`

Cancel out terms in the numerator:
`f'(x) = limh->0 [(cancel(5x)+5hcancel(+1)cancel(-5x)cancel(-1))/h]`
`f'(x)=limh->0[(5cancel(h))/cancel(h)]`
`f'(x)=5`