Derivative of a differential equation
I'm following a paper where at some point the author get the following derivative:
$$frac{partial V}{partial L}=Y-Xfrac{dY}{dX}=alpha X^{-frac{c}{b}}Y^{frac{1}{b}}$$
where: $Y=frac{V}{L}$ and $X=frac{K}{L}$
Then, starting from this he calculates the partial derivative with respect
to L and the cross second-order partial derivative (the partial derivative with respect to K), whose results are shown below:
$$frac{partial^2 V}{L^{2}}=-frac{alpha }{bL} X^{-frac{c}{b}}Y^{frac{1}{b}-1}left ( Xfrac{dY}{dX}-cY right )$$
$$frac{partial^2 V}{dKdL}=frac{alpha }{bL} X^{-frac{c}{b}-1}Y^{frac{1}{b}-1}left ( Xfrac{dY}{dX}-cY right )$$
Since this is the first time that I have to work with differential equations, I would like some advice on how to proceed to obtain the results mentioned above. Sorry if the question is stupid (I am an economist and not a mathematician :)) and every help/hint or suggestion will be very valuable.
Thank you so much,
Alessandro
differential-equations derivatives partial-derivative
asked Nov 26 at 13:10
Alessandro
205
add a comment |
I'm following a paper where at some point the author get the following derivative:
$$frac{partial V}{partial L}=Y-Xfrac{dY}{dX}=alpha X^{-frac{c}{b}}Y^{frac{1}{b}}$$
where: $Y=frac{V}{L}$ and $X=frac{K}{L}$
Then, starting from this he calculates the partial derivative with respect
to L and the cross second-order partial derivative (the partial derivative with respect to K), whose results are shown below:
$$frac{partial^2 V}{L^{2}}=-frac{alpha }{bL} X^{-frac{c}{b}}Y^{frac{1}{b}-1}left ( Xfrac{dY}{dX}-cY right )$$
$$frac{partial^2 V}{dKdL}=frac{alpha }{bL} X^{-frac{c}{b}-1}Y^{frac{1}{b}-1}left ( Xfrac{dY}{dX}-cY right )$$
Since this is the first time that I have to work with differential equations, I would like some advice on how to proceed to obtain the results mentioned above. Sorry if the question is stupid (I am an economist and not a mathematician :)) and every help/hint or suggestion will be very valuable.
Thank you so much,
Alessandro
differential-equations derivatives partial-derivative
asked Nov 26 at 13:10
Alessandro
205
add a comment |
I'm following a paper where at some point the author get the following derivative:
$$frac{partial V}{partial L}=Y-Xfrac{dY}{dX}=alpha X^{-frac{c}{b}}Y^{frac{1}{b}}$$
where: $Y=frac{V}{L}$ and $X=frac{K}{L}$
Then, starting from this he calculates the partial derivative with respect
to L and the cross second-order partial derivative (the partial derivative with respect to K), whose results are shown below:
$$frac{partial^2 V}{L^{2}}=-frac{alpha }{bL} X^{-frac{c}{b}}Y^{frac{1}{b}-1}left ( Xfrac{dY}{dX}-cY right )$$
$$frac{partial^2 V}{dKdL}=frac{alpha }{bL} X^{-frac{c}{b}-1}Y^{frac{1}{b}-1}left ( Xfrac{dY}{dX}-cY right )$$
Since this is the first time that I have to work with differential equations, I would like some advice on how to proceed to obtain the results mentioned above. Sorry if the question is stupid (I am an economist and not a mathematician :)) and every help/hint or suggestion will be very valuable.
Thank you so much,
Alessandro
differential-equations derivatives partial-derivative
asked Nov 26 at 13:10
Alessandro
205
I'm following a paper where at some point the author get the following derivative:
$$frac{partial V}{partial L}=Y-Xfrac{dY}{dX}=alpha X^{-frac{c}{b}}Y^{frac{1}{b}}$$
where: $Y=frac{V}{L}$ and $X=frac{K}{L}$
Then, starting from this he calculates the partial derivative with respect
to L and the cross second-order partial derivative (the partial derivative with respect to K), whose results are shown below:
$$frac{partial^2 V}{L^{2}}=-frac{alpha }{bL} X^{-frac{c}{b}}Y^{frac{1}{b}-1}left ( Xfrac{dY}{dX}-cY right )$$
$$frac{partial^2 V}{dKdL}=frac{alpha }{bL} X^{-frac{c}{b}-1}Y^{frac{1}{b}-1}left ( Xfrac{dY}{dX}-cY right )$$
Since this is the first time that I have to work with differential equations, I would like some advice on how to proceed to obtain the results mentioned above. Sorry if the question is stupid (I am an economist and not a mathematician :)) and every help/hint or suggestion will be very valuable.
Thank you so much,
Alessandro
differential-equations derivatives partial-derivative
differential-equations derivatives partial-derivative
asked Nov 26 at 13:10
Alessandro
205
asked Nov 26 at 13:10
Alessandro
205
asked Nov 26 at 13:10
Alessandro
205
asked Nov 26 at 13:10
Alessandro
205
asked Nov 26 at 13:10
Alessandro
205
205
add a comment |
add a comment |
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