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!

rp1 said:

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!

Click to expand...

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.

\(\displaystyle \dfrac{ \delta x}{ \delta y} = \dfrac{g - k}{k(y - g)^2}.\)

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?

thanks - all clear

thanks - all clear