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Math Help - Differentiation of an integral with implicitly defined variable

  1. #1
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    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

    Differentiation of an integral with implicitly defined variable-formula-1.jpg

    while ^z_i is implicitly defined by the following equation

    Differentiation of an integral with implicitly defined variable-formula-2.jpg

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

    Differentiation of an integral with implicitly defined variable-formula-3.jpg

    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
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  2. #2
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    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: 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 q_i. In other words, you want to see how V_i changes when q_i changes and all the other variables are held constant.

    \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,

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

    \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, D_i is a constant, so the derivative of a function minus a constant is just the derivative of that function:

    \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, V_i^i=\frac{\partial{V^i}}{\partial{q_i}} and 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:

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

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

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

    Cheers,
    Peter
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