# Percentage variation dependance of a function of two variables

The language is a sort of barrier in this case (even in my native language) so I'll try to make an example here to clarify the question.

Given a function $f(a,b)$ I want to answer the question: to which variable is more "sensitive" this function? I mean, if I change by $2\%$ each variable at a time, which one will produce the greatest percentage change in $f(x)$?

The function $f(x)$ is the one that describes Fraunhofer's diffraction of a light beam on a screen far from the source. Being $w$ the width of the slit through wich the beam is diffracted, $\ell$ the distance of the screen from the slit and $\lambda$ the light wavelenght

$$I(x) = I_0\frac{\sin^2\beta^2}{\beta^2}$$ where $$\beta = \frac{\pi w}{\ell\lambda}x$$

Even if the equation is not strictly this one, I am interested in this form in particular and want to know if $I(x)$ varies more by changing $w$ or $\ell$ and be able to give a quantitive information.

Thank you all in advance and I hope I made my problem clear.

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You want whichever variable has the property that the partial derivative with respect to it is larger in absolute value at a given point. –  Qiaochu Yuan May 26 '12 at 13:55
I can't answer on a point-by-point basis. Since I need to match this model to experimental data, I need a broader result. Absolute values are not useful, given that $w$ is at least three orders of magnitude smaller than $\ell$ ($w \approx 10^{-4}m$ versus $d \approx 10^0 m$) –  Marcello Massaro May 26 '12 at 14:07
There are various ways to integrate this information over all points and they are not equivalent. What do you want to do this for? –  Qiaochu Yuan May 26 '12 at 14:20
Since both parameters ($w$ and $\ell$) are measured data, I want to know which one is to be measured more carefully. –  Marcello Massaro May 26 '12 at 14:26
Do you expect there to be orders of magnitude of difference between the sensitivities? If so, I guess you could do something like integrate the square of the partial derivatives with respect to each variable over the relevant space of parameters. –  Qiaochu Yuan May 26 '12 at 14:28