Question

How do you use the definition of a derivative to find the derivative of `f(x) = 5x + 9` at x=2?

Answer 1

`f'(2) = 5`

Explanation:

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

So, with `f(x) = 5x+9` we have:

`f'(2) = lim_(hrarr0)( ( f(2+h) - f(2) ) / h )`
`:. f'(2) = lim_(hrarr0)( ( (5(2+h)+9) - (5(2)+9) ) / h )`
`:. f'(2) = lim_(hrarr0)( ( (10+5h+9) - (10+9) ) / h )`
`:. f'(2) = lim_(hrarr0)( ( 10+5h+9 - 10-9 ) / h )`
`:. f'(2) = lim_(hrarr0)( (5h) / h )`
`:. f'(2) = lim_(hrarr0)( 5 )`
`:. f'(2) = 5`