Derivative of $f(x) = (x+x)$ I'm trying to teach myself algebra and derivatives. I learned the derivative for $f(x) = x^2$ from a lesson, and now I thought I would see if I could figure out the derivative of $f(x) = x+x$ on my own.  
I know the formula for derivatives is: $$\lim_{\delta\rightarrow 0}\frac{f(x+\delta) 
        - f(x)}{\delta}$$
So my attempt at algebra amounted to this: $$\frac{((x+x) + (\delta + \delta)) 
        - (x+x)}{\delta}$$
$$=\frac{\delta + \delta}{\delta}$$
Which doesn't seem right. (Isn't the derivative supposed to NOT contain the delta term?)
 A: Add the $\delta$s and simplify, you get $\displaystyle \frac{2\delta}{\delta} = 2$. The derivative is then $2$.
A: Here is another great way to calculate the derivative of $f(x)=(x+x)$, instead of using the Limit Definition, which is the method you are using above. 
The other way to do this is to use a combination of methods called the Power Rule, and Sum Rule.
Here is a mathematical representation of the Sum Rule:
$$(f + g)'(x) = f'(x) + g'(x)$$
This says $(f + g)$ prime, where prime means "the derivative of."
So this rule literally states: the derivative of $(f + g)(x)$ is equal to the derivative of $f$, plus the derivative of $g$.
In your example this would break down to:
$$f'(x)= (x+x)' = x' + x'$$
Now we will use the Power Rule to calculate the derivative of each individual piece (the $x's$ in this case).
The Power Rule states:
$$\frac{d}{dx} x^n = nx^{n-1} $$
Note: $\displaystyle \frac{d}{dx}$ means to take the derivative with respect to $x$. So this rule states that the derivative of $x^n$ is equal to multiplying $x$ by $n$,and then subtracting $1$ from the exponent. 
Here is the Power rule applied to $x' + x'$:
Remember that $x$ is really $x^1$, so
$$\frac d{dx}(x^1) = 1\cdot x^{1-1} = x^0 = 1$$
Applying this rule to both $x's$ we end up with 
$$1 + 1 = 2$$
Which is your same result. Hopefully this made sense you. If you get the hang of these rules they make calculating your derivatives super quick, and also provide you with a way to check your Limit Definition calculation. Good luck!!
