Derivative with a summation operator?

[Elesirdur]

Edit: Turns out my original formula had some errors, updated and correct one is shown below:

Quick question for the math/stats/science/econ folks out there. Say I have a function that includes a summation operator, like the one below. For the sake of this problem, let alpha be some scalar constant.



Obviously, if the summation operator was not there the derivative w.r.t. x would be 6x+alpha. I don't believe I ever learned the rules for taking the derivative which includes the summation operator, and some googling hasn't turned much up.

This kind of problem is starting to pop up now that I'm having to deal with structural equation models in my econ classes, but I'm not sure how to proceed when trying to determine the relationship between alpha (likely a OLS estimator) and x.

Thanks for any responses.

[Osede]

if x is independent of x_i, then the partial derivative w.r.t x_i is only the alpha

[Quicklet]

Generally, if you're dealing with well-behaved functions the derivative of a summation is equal to the summation of the derivatives of each term (i.e. you can switch the order of the two, which you can see just by checking a few basic examples).

[Elesirdur]

Damnit, also realised that I forgot the i subscript on the x in the first term.

Updated OP, but here's what it should look like as well:

[BerenTebogo]

I'd think it's just (6x + alpha) then, really

Err, derp, fixed

[Drex]

Damnit, also realised that I forgot the i subscript on the x in the first term.

Updated OP, but here's what it should look like as well:

If you mentally extend the summation into its terms, the derivative is just 6x + a + a + a + a... n times, or 6x+a*n.

No?

[Quicklet]

Keep in mind you still have to evaluate the summation after you take the derivative.

[Blubbartron]

Drex is correct. The derivative of a summation on some polynomial is the summation of the derivative on the polynomial.

[Woozie]

My entire post assumes there's supposed to be a parenthesis in your equation.

To me it looks like 6x+alpha. Since the operator depends on i, you can't just move it inside the summation over i unless your class uses some notation/conventions I'm not familiar with.

http://latex.codecogs.com/gif.latex?...n}\alpha%20x_n

As you can see, moving the operator into the summation changes the answer entirely and hence, you cannot just move things in and out of a sum if they depend on the index of summation. It's the same as how you can't move things in and out of an integral if they depend on the variable of integration.

http://latex.codecogs.com/gif.latex?...t%20f(x)g(x)dx

If you want to move things into an integral or sum, you have to change the index so that it doesn't match any index of what you're putting into the sum

http://latex.codecogs.com/gif.latex?...a%20f(x)g(t)dt

This works because I changed the dummy variable. Sums follow the same rules.

[Woozie]

Drex is correct. The derivative of a summation on some polynomial is the summation of the derivative on the polynomial.

I think I see why there's some confusion here. What Elesirdur posted is not a polynomial.

http://latex.codecogs.com/gif.latex?...1}\alpha_i%20x

is a polynomial. Elesirdur has is

http://latex.codecogs.com/gif.latex?...1}\alpha%20x_i

which is a function of n variables, and he's trying to take the partial derivative with respect to the ith variable.

Or maybe I'm missing something here.

[Blubbartron]

Good catch Woozie, as usual. I didn't even notice the subscript.

[Quicklet]

Using i for both the index of summation and then for variables outside the summation is ambiguous notation at that. Better to use a different letter for the index of each summation you have in an equation (like j, or if you had 2 separate summations you could use j for one and k for the other), and then use a different letter (like i) for your "floating" index that isn't part of any summation.

Then you can use the rule that the partial derivative of x_j with respect to x_i is a Kronecker delta which will pick out the appropriate term from the sum. I don't know the level of math your course is going to get to with this type of thing, but if it does get more complex, keeping your index notation clear will probably save you from making some mistakes down the road.