Forgot to say one thing more --- this time I want a reply from you (xpi0t0s) about who is right ....

... I know you would try to find out who actually is.

... I know you would try to find out who actually is.

... I know you would try to find out who actually is.

Two questions follow from that.

(1) Do you agree that for all x, f'(x)=5?

(2) Do you agree that for x=5, f'(x)=5?

I anticipate Yes for both, but if at either point you say no, please explain why.

If f(x) = x + x + .. (x times).. + x, then answer would be --

(1) No

(2) No

Explanation :

(1) For all x, f'(x) would not be some constant like 5, or 10.

It would be 2x.

(2) For x = 5, f'(x) would be 2*5 = 10.

(Very basic differentiation question.)

(1) Yes

(2) Yes

because f'(x) = 5 which is independent of x.

Yeah.

f(x) = 5x => f'(x) = 5 = x.

So ???

f(x) = 5x => f'(x) = 5 = x.

So ???

So, generalising this, if f(x)=x+x+x+... (N times) and x=N, do you agree or disagree that f'(x)=N?

Yeah .. as long as N is a number like 2, 3 -9, 11.2 or any number ... f'(x) = N.

OK, so since f'(x)=N and N=x, f'(x) must equal x. QED.

f'(x) = x as long as N is a number independent of x.

When N is independent of x, f'(x) = N.

See, f(x) = x + x + ..(N times).. + x = Nx

So, f'(x) = N.

I think an example might make things a bit clearer.

Let us define f(x) = x + x + x = 3x.

So f'(x) = 3.

Now, f'(x) = 3 is completely independent of x.

So, f'(x) is not gonna change whether x=3,4,5 or anything.

Universally, if f(x) = 3x, f'(x) is always equal to 3, whatever x be.

D(Nx) = N. But, D(x.x) != x.

Tell me one thing ... if f(x) = 5x,

what is f'(x) when x = 100 ? 5 right ??

and what about x = 200 ? again 5 right ??

So, can you say 100 = 200 ?? NO.

When you write f(x) = Nx, your func is a linear one with constant slope -- so has a constant derivative at all values of x.

When you write f(x) = x times x, it depends on x twice. So, it is basically quadratic in nature. It's slope varies with x.

You are accepting things in the wrong way .. I mean try to feel the Maths behind, not the formula.

Formulas may deceive you, but logic never does.

And, if my explanation still doesn't satisfy you.. why not ask someone who you trust ??