Find f(x+ (delta)x)- f(x)

---------------------
(delta)x

if f(x)=8x(squared)+ 1

This is so confusing! What is all the delta x stuff?

delta x is a small change in x.

What you are doing is drawing a short line from (x,y) to (x+delta x, y at x + delta x)
Then you are finding the slope of that line
the derivative of f(x) is the limit of the slope of that line as delta x goes to zero.
f(x+delta x) = 8 (x+delta x)^2 +1
f(x) = 8 x^2 +1
so
f(x+delta x) = 8 x^2 +16 x delta x + 8 delta x^2 +1
subtract f(x) = 8 x^2 + 1 from that
then
f(x+delta x)-f(x) = 16 x delta x + 8 delta x^2
now divide by delta x
[f(x+delta x)-f(x)]/delta x =

16 x + 8 delta x

that is the answer to your question

going one step beyond, look what happens when delta x goes to zero
the derivative then f'(x) = 16 x

Please tell me if you did not follow this and I will try again some other way. It is really, really important.

delta x is a small change in x. I'm just going to call it dx. dx/x <<1

For this particular f(x) funtion,

[f(x + dx)f f(x)]/dx
= [8*(x + dx)^2 +1 - 8*x^2-1]/dx
= [16x*dx + 8(dx)^2]/dx
= 16x + 8*dx

As dx becomes infinitiely small compared to x, the above expression becomes 16x and is known as the derivative of the function.

http://www.sosmath.com/calculus/diff/der00/der00.html

I understand that the notation with delta x can be confusing at first. Delta x (Δx) represents a small change in the variable x. In this problem, we are given a function f(x) and are asked to find a difference quotient, which measures the average rate of change of the function f(x) between two points x and x + Δx.

To find f(x + Δx), we substitute (x + Δx) into the expression for the function f(x). In this case, f(x) = 8x^2 + 1, so we have:
f(x + Δx) = 8(x + Δx)^2 + 1

Expanding the expression (x + Δx)^2, we get:
f(x + Δx) = 8(x^2 + 2xΔx + (Δx)^2) + 1
= 8x^2 + 16xΔx + 8(Δx)^2 + 1

Now, to find the difference f(x + Δx) - f(x), we subtract the expression for f(x) from the expression for f(x + Δx):
f(x + Δx) - f(x) = (8x^2 + 16xΔx + 8(Δx)^2 + 1) - (8x^2 + 1)
= 16xΔx + 8(Δx)^2

Finally, we divide the difference by Δx to get the average rate of change:
(f(x + Δx) - f(x)) / Δx = (16xΔx + 8(Δx)^2) / Δx
= 16x + 8Δx

So, the expression (f(x + Δx) - f(x)) / Δx simplifies to 16x + 8Δx.