Question

How do you differentiate `f(x) = 3`?

Answer 1

Using the definition?

there's nothing very special about `3` here.

`f(x)=c` (Note this makes `f(x+h)=c` as well.

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

`=lim_(hrarr0)(c-c)/h = lim_(hrarr0)0/h`

Fod `h!=0`, we have `0/h=0` so, we continue:

`= lim_(hrarr0)0 = 0`

That is: for `f(x) = c`, we have `f'(x) = 0`.

Note that this makes sense geometrically too. We "get at" the slope of the tangent line by looking at slopes of secant line to the graph and then finding a limit.

For any two points on the graph of `f(x)=3`, we have slope `m = 0` Taking a limit wont't change this slope. (Next exercise, think about the slope of any line `y = mx+b`.)