# Is it true If the graph of the function y=f(x) is a line, then the functions ∆y and dy for f(x) at x=3 are identical? if so why and if not what is a counter example?

sciencesolve | Certified Educator

You need to consider the equation of a linear function `y = ax + b` and you need to find derivative of the function at a given point `x = 3,` such that:

`(Delta y)/(Delta x) = (f(x+Delta) - f(x))/(Delta x) => (Delta y)/(Delta x)(ax + aDelta x + b - ax - b)/(Delta x) = a `

` `

`(dy)/(dx)|_(x = 3) = f'(3) = lim_(x->3) (f(x) - f(3))/(x - 3)`

`f'(3) = lim_(x->3) (ax + b - 3a - b)/(x - 3)`

Reducing duplicate members yields:

`f'(3) = lim_(x->3) (ax - 3a)/(x - 3)`

Factoring out a to numerator yields:

`f'(3) = lim_(x->3) a(x - 3)/(x - 3)`

Reducing duplicate factors yields:

`f'(3) = lim_(x->3) a = a`

Hence, evaluating `(dy)/(dx)|_(x = 3)` yields that `(dy)/(dx)|_(x = 3) = (Delta y)/(Delta x)|_(x = 3) = a` , thus `(dy)/(dx)|_(x = 3)` and `(Delta y)/(Delta x)|_(x = 3)` for a linear function `y = ax + b` are identical.