The derivative of a function is said to be $\lim\_{\Delta{x}\to 0} \frac{f(\Delta{x}+x)-f(x)}{\Delta{x}}$. This I believe I understand. But, I always have trouble believing something that is so theoretical without walking through a concrete example. So, I took the derivative of the function $f(x)=x^2+2x+4$ and I, of course, got $f'(x)=2x+2$. So, this says that for a marginal change in $x$, there will be a change in $y$ of $2(1)+2=4$.

I evaluate the original function at $x=1$ and I find that $y=7$. Then I evaluate the original function at $x=2$ and I find that $y=12$. In this case $\frac{\Delta{y}}{\Delta{x}}=5$, not the $4$ predicted by our derivative function. Where is the mistake in my thinking? Can someone provide me with a concrete example of the derivative that will show me that it does work?