How do I approximate this correctly? X=(A+B)/(C+D) X´=(A+Delta A + B + Delta B)/(C+ D

[hotzenplotz515]

How do I approximate this correctly? X=(A+B)/(C+D) X´=(A+Delta A + B + Delta B)/(C+ D

Hey,

X=(A+B)/(C+D)
X´=(A+Delta A + B + Delta B)/(C+ Delta C + D+ Delta D)
Y=X´- X

I want to calculate how much of the value of Y is explained by Delta A, Delta B, Delta C and Delta D. What is the impact of each Delta on Y.

Greets

 

[HallsofIvy]

So \(\displaystyle y= \frac{A+ \Delta A+ B+ \Delta B}{C+ \Delta C+ D+ \Delta D}- \frac{A+ B}{C+ D}\).

I'm not certain what you mean by "is explained by" but I think you mean how much a change in any one of the "Deltas" changes the value of y. That is the "partial derivative with respect to that quantity.

That would be
\(\displaystyle \frac{\partial y}{\partial \Delta A}= \frac{1}{C+ \Delta C+ D+ \Delta D}\)
\(\displaystyle \frac{\partial y}{\partial \Delta B}= \frac{1}{C+ \Delta C+ D+ \Delta D}\)
\(\displaystyle \frac{\partial y}{\partial \Delta C}= \frac{A+ \Delta A+ B+ \Delta B}{(C+ \Delta C+ D+ \Delta D)^2}\)
\(\displaystyle \frac{\partial y}{\partial \Delta D}= \frac{A+ \Delta A+ B+ \Delta B}{(C+ \Delta C+ D+ \Delta D)^2}\)