Hi, I have unfortunately forgotten my calculus.... I have a problem regarding partial vs. total derivatives. I have a function f(x,y) = ax - by (this is not the actual function, but for the sake of argument let's say it looks like this). Also, x and y are not independent: x = c - (y-1)^2.
My goal is to find the ratio of sensitivities of f with respect to x and y. In other words, if I want f to remain constant, and I know x is going to move by some amount, how much should y move in order for that to happen?
I have a few thoughts about how to compute this but am not sure which is most appropriate/accurate/etc.
These are the ways I am considering:
1) Compute partial derivatives of f w.r.t. x and y and take the ratio. I think this is wrong because of the inter-dependence of x and y.
2) Compute partial derivatives of f w.r.t. x and y, but assume that x has been constrained by y, in other words that x is dependent upon y and not vice-versa. Substitute the equation for x into the original function and compute del f / del x. However then I am not sure how to compute del f / del y.
3) Compute a total derivative of f w.r.t. x and y. E.G.
df / dx = del f / del x + (del f / del y) * (del y / del x)
where del means partial derivative. Similar equation for df / dy. Then take the ratios of those. The question then is how to compute del x / del y and del y / del x. Do I just solve for one in terms of the other, using the second relation given, and differentiate the results?
OR - 4) none of the above?
All help very greatly appreciated and thanks in advance!!