# Thread: Derivative Functions - Is This a Proper Proof?

1. ## Derivative Functions - Is This a Proper Proof?

I know that f(x) is derivative at x=0, and I need to prove the following, while x<a<y :

Is this a proper proof? Thank you very much!

2. Originally Posted by adam63
I know that f(x) is derivative at x=0, and I need to prove the following, while x<a<y :

Is this a proper proof? Thank you very much!

What is this? What is f(x)? There're lots of functions which are derivable at some point but not at another point...

Tonio

3. Originally Posted by tonio
What is this? What is f(x)? There're lots of functions which are derivable at some point but not at another point...

Tonio
I need to prove it the way it is :
let f(x) be derivative at x=a, and I need to prove that:
$\displaystyle lim\frac{f(x)-f(y)}{x-y} = f'(a)$
(while $\displaystyle x,y \rightarrow a ; x<a<y$)

4. Look at $\displaystyle \frac{f(x)- f(y)}{x- y}= \frac{f(x)- f(a)}{x- y}- \frac{f(y)- f(a)}{x- y}$$\displaystyle =\left(\frac{x-a}{x- y}\right)\frac{f(x)- f(a)}{x- a}-\left(\frac{y-a}{x-y}\right)\frac{f(y)- f(a)}{y- a} 5. Originally Posted by HallsofIvy Look at \displaystyle \frac{f(x)- f(y)}{x- y}= \frac{f(x)- f(a)}{x- y}- \frac{f(y)- f(a)}{x- y}$$\displaystyle =\left(\frac{x-a}{x- y}\right)\frac{f(x)- f(a)}{x- a}-\left(\frac{y-a}{x-y}\right)\frac{f(y)- f(a)}{y- a}$
Thank you for your suggestion!

Is my way wrong? If so, where is it wrong (I'm a 'calculus beginner', so I will be glad to know where I should fix myself)

Thank you

6. And by the way, I didn't understand how to continue your suggestion for a proof, Tonio. Where does the part of $\displaystyle x,y \rightarrow$ a appears?

Thank you very much!

7. Originally Posted by adam63
Thank you for your suggestion!

Is my way wrong? If so, where is it wrong (I'm a 'calculus beginner', so I will be glad to know where I should fix myself)

Thank you
The problem is that you don't show how to go from one step to the next. The next to last line is $\displaystyle \lim_{x,y\to a}\frac{f(y)- f(x)}{y- x}$ and the line just before that is $\displaystyle \lim_{x,y\to a}\frac{f(x+ y- a)- f(x)}{y- x}$. How did you get from one to the other?

8. That's because:

$\displaystyle x \rightarrow a, then (x-a) \rightarrow 0.$