Find and simplify the difference quotient [f(x+h)-f(x)]/h for the given function.

f(x)=3x^2+2x+1

Question

Find and simplify the difference quotient [f(x+h)-f(x)]/h  for the given function.

f(x)=3x^2+2x+1

anonymous · Answer

Do you have the answer just so I can see if I'm correct?

anonymous · Answer

I do, however I think the answer is incorrect. That is why I am asking this question. 

The supposed answer is:
$$6x+3h+2$$

nincompoop · Answer

how did you get that answer @anthonykanow

anonymous · Answer

That's what I got, I'll explain the steps

nincompoop · Answer

that is not the supposed answer 
use the power rule

nincompoop · Answer

X^n = nX^n-1+...

nincompoop · Answer

the derivative of a constant is zero

anonymous · Answer

@nincompoop I did not get that answer, my professor did. 
Also, this is not suppose to be solved using calculus.

nincompoop · Answer

[f(x+h)-f(x)]/h <-- definition of derivative (it is slightly incomplete, but it is known as the "difference quotient")

plug your function into the definition and show it here

anonymous · Answer

So when we are looking at a difference quotient problem we have to substitute the function into the formula we are given a couple of times, and we have to simplify. 

When we substitute,
we get ((3(x+h)^2+2(x+h)+1)-(3x^2+2x+1))/h
This simplifies to
((3(x^2+2xh+h^2)+2x+2h+1)-3x^2-2x-1)/h
when we multiply out the three, we can see that many terms will cancel

this leaves us with (6xh+3h^2+2h)/h
and when we cancel the available h in each term we are left with 3h+6x+2

anonymous · Answer

@nincompoop What answer did you arrive to?
I do not think you are correct. This is a simple algebra problem.

anonymous · Answer

@HiGhoCtaN399 Thank you very much.
I have found my error.

nincompoop · Answer

what is the purpose of adding h in that equation?

anonymous · Answer

I think it basically has to do with rates of change and limits and stuff but I'm not entirely sure what purpose the h serves.

nincompoop · Answer

difference quotient computes for the slope of a secant line
the reason I brought up the power rule, because that is the basis of the limit (where h approaches zero but not equal to zero) 
you can't compute for the slope of the secant line if you are not given the points in a typical algebraic manipulation. it only gives you the general form.