# Mean theorim question

1. Nov 21, 2008

### transgalactic

2. Nov 21, 2008

### Unco

If h(x) = f(x) for all x then h'(x) = f'(x).

Last edited: Nov 21, 2008
3. Nov 21, 2008

### transgalactic

for a fact
??

4. Nov 21, 2008

### Unco

Yes; consider the definition $$h'(x) = \lim_{t\to 0}\frac{h(x+t)-h(x)}{t}$$.

5. Nov 21, 2008

### Pere Callahan

If f(x)=g(x) for all x, then the two functions are the same. And for this very reason they have the same derivative.

6. Nov 21, 2008

### transgalactic

7. Nov 21, 2008

### Unco

Now you're just confused by notation. Let's use some neutral letters.

If A and B are differentiable functions on [a,b] such that A(x) = B(x) for all x in [a,b], then A'(x) = B'(x).

Why? $$A'(x) = \lim_{t\to 0}\frac{A(x+t)-A(x)}{t} = \lim_{t\to 0}\frac{B(x+t)-B(x)}{t} = B'(x)$$.

Also, they said "h'(c)=f'(c)-g'(c)" NOT "h'(c)=f'(x)-g'(x)".

8. Nov 21, 2008

### transgalactic

when you say "A(x) = B(x) for all x in [a,b]"

that mean that they share each one of their points
which means that A(x) and B(x) are the same function then of course A'(x) = B'(x) (its the same function)

which is not true in the graph

Last edited: Nov 21, 2008
9. Nov 21, 2008

### Pere Callahan

Take A(x)=h(x) and B(x)=f(x)-g(x)

10. Nov 21, 2008

ahhhh thanks