Thread: Differentiation of an integral with implicitly defined variable

1. 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

while ^z_i is implicitly defined by the following equation

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

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

2. 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: $\displaystyle V^i=\int_{\hat{z}_i}^{z} (R^i(q_i,q_j,z_i)-D_i) f(z_i) dz_i$

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

$\displaystyle \frac{\partial{V^i}}{\partial{q_i}} = \frac{\partial}{\partial{q_i}} \int_{\hat{z}_i}^{z} (R^i(q_i,q_j,z_i)-D_i) f(z_i) dz_i$

Assuming you can exchange the derivative with the integral,

$\displaystyle \frac{\partial{V^i}}{\partial{q_i}} = \int_{\hat{z}_i}^{z} \frac{\partial}{\partial{q_i}} $(R^i(q_i,q_j,z_i)-D_i) f(z_i)$ dz_i$

Since $\displaystyle f(z_i)$ doesn't vary with respect to $\displaystyle q_i$, we can treat it like a constant. The derivative of a constant times a function is the constant times the derivative:

$\displaystyle \frac{\partial{V^i}}{\partial{q_i}} = \int_{\hat{z}_i}^{z} \frac{\partial}{\partial{q_i}} $(R^i(q_i,q_j,z_i)-D_i)$ f(z_i) dz_i$

And lastly, $\displaystyle D_i$ is a constant, so the derivative of a function minus a constant is just the derivative of that function:

$\displaystyle \frac{\partial{V^i}}{\partial{q_i}} = \int_{\hat{z}_i}^{z} \frac{\partial{R^i(q_i,q_j,z_i)}}{\partial{q_i}} f(z_i) dz_i$

And I assume that in your notation, $\displaystyle V_i^i=\frac{\partial{V^i}}{\partial{q_i}}$ and $\displaystyle R_i^i(q_i,q_j,z_i)=\frac{\partial{R^i(q_i,q_j,z_i) }}{\partial{q_i}}$, so the result is:

$\displaystyle V_i^i = \int_{\hat{z}_i}^{z} R_i^i(q_i,q_j,z_i) f(z_i) dz_i$

- Hollywood

3. Re: Differentiation of an integral with implicitly defined variable

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

Cheers,
Peter