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Thread: Derivative of derivative? if dx/dy = (g - k)/(k*(y-g)^2)...

  1. #1
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    Derivative of derivative? if dx/dy = (g - k)/(k*(y-g)^2)...

    Hi,

    I am trying to work out how a derivative (sensitivity of x to variable y) is affected by the change in a variable g and variable k.

    if dx/dy = (g - k)/(k*(y-g)^2)

    I know this is negative for K>g

    The derivative of dx/dy with respect to k is:

    d (dx/dy) / dk = -g/(k^2*(y-g)^2)

    This is negative. What does the negative sign mean? That sensitivity to y is lower if k is higher (or the opposite)?

    The derivative of dx/dy with with respect to g is more difficult. I think it is (correct me if I'm wrong):

    d (dx/dy) / dg = [1 / (k*(y-g)^2)]*[1-2(k-g)/(y-g)]

    I think this is negative if:

    (k-g) / (y - g) > 0.5

    Again, if it is negative, what does that mean exactly for sensitivity of x to y if g is higher?

    Many thanks for any help on this!

  2. #2
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    Quote Originally Posted by rp1 View Post
    Hi,

    I am trying to work out how a derivative (sensitivity of x to variable y) is affected by the change in a variable g and variable k.

    if dx/dy = (g - k)/(k*(y-g)^2)

    I know this is negative for K>g

    The derivative of dx/dy with respect to k is:

    d (dx/dy) / dk = -g/(k^2*(y-g)^2)

    This is negative. What does the negative sign mean? That sensitivity to y is lower if k is higher (or the opposite)?

    The derivative of dx/dy with with respect to g is more difficult. I think it is (correct me if I'm wrong):

    d (dx/dy) / dg = [1 / (k*(y-g)^2)]*[1-2(k-g)/(y-g)]

    I think this is negative if:

    (k-g) / (y - g) > 0.5

    Again, if it is negative, what does that mean exactly for sensitivity of x to y if g is higher?

    Many thanks for any help on this!
    Why is this posted in algebra when it seems to be a question in multivariate calculus?

    Although it makes no mathematical difference, it is usual to make y the dependent variable when using x and y, and using common conventions helps with communication. Furthermore, you seem to be talking about partial derivatives, not derivatives, which adds a small extra element of confusion to your post. It might help if you give the original function: as it is, we do not know for sure if anything in your post is correct.

    [tex]\dfrac{ \delta x}{ \delta y} = \dfrac{g - k}{k(y - g)^2}.[/tex]

    Assuming the above is correct, you can deduce the following qualitative conclusions from that alone.

    The greater | y - g | is, the smaller is the sensitivity of x to changes in y.

    If g > k > 0, then the sensitivity of x to changes in y is positive (meaning an increase in y causes an increase in x whereas a decrease in y causes a decrease in x.)

    If 0 < g < k, then the sensitivity of x to y is negative.

    If g = k > 0, x is not sensitive at all to changes in y.

    Things get a bit more complicated if k is negative.

    The greater that | g - k | is, the greater is the degree of sensitivity of x to changes in y.

    The smaller that | k | is, the greater is the degree of sensitivity of x to changes in y.

    Is that what you wanted to know?
    Last edited by JeffM; 03-09-2018 at 09:32 AM.

  3. #3
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    thanks - all clear

    thanks - all clear

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