# Thread: Simple question -definition of derivative

1. ## Simple question -definition of derivative

We've all seen the definition, but for some reason I still don't understand the logic behind it. This is the part of the definition that I have a problem with:

f (x+h) - f (x) / (x+h) - x

I understand that (x+h) - x = h.

If this is the case, will this be true f (x+h) - f (x) = h ...?

Thus, h / h ?

2. Hi there,

This part is given

$\frac{f (x+h) - f (x)}{(x+h) - x}$

then as you have shown there is some cancellation in the denominator

$\frac{f (x+h) - f (x)}{h}$

This is where it stops. No more work needs to be done.

If you want to find a derivative from here find the limit of your function as $h \to 0$ using the deifnition above.

3. Originally Posted by pinkperil
We've all seen the definition, but for some reason I still don't understand the logic behind it.
Do you understand how to determinte the slope of the chord between the two points $\left(x,f(x)\right)~\&~\left(x+h,f(x+h)\right)?$
That is the logic of it all.

4. Originally Posted by pinkperil
We've all seen the definition, but for some reason I still don't understand the logic behind it. This is the part of the definition that I have a problem with:

f (x+h) - f (x) / (x+h) - x

I understand that (x+h) - x = h.

If this is the case, will this be true f (x+h) - f (x) = h ...?
NO, f(x+ h) is NOT equal to f(x)+ h! Exactly what it is depends on the function f.

Thus, h / h ?
Well, that would make calculus pretty trivial wouldn't it?

If f(x) is a linear function, f(x)= ax+ b, then f(x+h)= a(x+h)+ b= ax+ah+ b.
f(x+h)= ax+ah+ b- ax+b= ah. (f(x+h)- f(x))/h= ah/h= a, the slope of the line.

If $f(x)= ax^2+ bx+ c$, then $f(x+h)= a(x+h)^2+ b(x+h+ c$ $= ax^2+ 2axh+ h^2+ bx+bh+ c$ so $f(x+h)- f(x)= ax^2+ 2axh+ h^2+ bx+ bh+ c- ax^2- bx- c= 2axh+ h^2+ bh$ and the "difference quotient" $\frac{f(x+h)- f(x)}{h}= \frac{2axh+ h^2+ bh}{h}= 2ax+ h+ b$

It gets really complicted for things like sin(x), cos(x), or log(x)!

5. Originally Posted by HallsofIvy
NO, f(x+ h) is NOT equal to f(x)+ h! Exactly what it is depends on the function f.
I see where my problem is. My problem is something even more fundamental: misunderstanding of functions. So if the function isn't what I assumed, then I must assume this:

y = f(x) = 9

f(x+h) - f(x) / h

9 - 9 / h

Although, it doesn't make much sense to me how f(x + h) = f(x) when this isn't true in many other cases like this:

f(x) = 2x
f(x + h) = 2x + 2h

Is it because the first situation has only a constant and the other has a variable? The reason why I thought f(x+h) - f(x) = h is because: (constant + h) - constant = h.

6. Originally Posted by pinkperil
I see where my problem is. My problem is something even more fundamental: misunderstanding of functions. So if the function isn't what I assumed, then I must assume this:

y = f(x) = 9

f(x+h) - f(x) / h

9 - 9 / h

Although, it doesn't make much sense to me how f(x + h) = f(x) when this isn't true in many other cases like this:
You quoted part of my post so you must have read it. How could you get "f(x+h)= f(x)" from it? f(x+h)= f(x) only for a constant function such as "f(x)= 9" where f(x)= f(y) no matter what x and y are.

And why would you say "if the function isn't what I assumed". You did not mention any particular function in your original post.

f(x) = 2x
f(x + h) = 2x + 2h

Is it because the first situation has only a constant and the other has a variable? The reason why I thought f(x+h) - f(x) = h is because: (constant + h) - constant = h.
If f is a constant, f(x)= c, then f(x+h)= c also so f(x+h)- f(x)= 0 and then (f(x+h)- f(x))/h= 0. Of course, the graph of a constant function is a horizontal straight line and has slope 0.

The function f(x)= x+ c has f(x+h)= x+ h+ c so f(x+ c)- f(x)= x+h+c- x- c= h and then (f(x+h)- f(x))/h= 1. The line y= x+ c has slope 1.

IF you were refering to specific function you should have said so in your first post. It appeared that your "f(x)" could be any function.

7. ...only for a constant function such as "f(x)= 9" where f(x)= f(y) no matter what x and y are.
This makes sense to me now. I couldn't wrap my head around (x + h) being something whole and my mind kept wanting to break it apart. Thanks a lot for your help. I appreciate it.