3 Attachment(s)

Differentiation of an integral with implicitly defined variable

Hey folks,

I am currently writing on a paper in economics and came up with the following problem I couldn't solve so far.

I would like to differentiate the integral from the following equation with regards to q_i

Attachment 25708

while ^z_i is implicitly defined by the following equation

Attachment 25709

The result is supposed to be the following equation, though I do not understand the way to arrive at it.

Attachment 25710

I would be glad if someone could help me to understand the way to the derivative. Please take into consideration that I have no background in mathematics :)

Cheers,

Peter

Re: Differentiation of an integral with implicitly defined variable

There are a lot of subscripts and functions that make this look harder than it really is.

You have:

Now you want to take the partial derivative with respect to . In other words, you want to see how changes when changes and all the other variables are held constant.

Assuming you can exchange the derivative with the integral,

Since doesn't vary with respect to , we can treat it like a constant. The derivative of a constant times a function is the constant times the derivative:

And lastly, is a constant, so the derivative of a function minus a constant is just the derivative of that function:

And I assume that in your notation, and , so the result is:

- Hollywood

Re: Differentiation of an integral with implicitly defined variable

Thanks a lot, Hollywood! You have been a great help to me.

Cheers,

Peter