Two questions about the propagation of uncertainty: $e^x$, and a complex iterative equation

I'm trying to do some calculations involving the propagation of uncertainties in complex equations as part of my PhD research, and I'm running into brick walls. One advisor isn't around and the other is the "ugh just go figure it out yourself" type. I even have a textbook, John R. Taylor "An Introduction to Error Analysis", which is greatly helpful except for the problems I'm presenting herein, which aren't addressed or aren't sufficiently discussed in the book.

The first is propagating uncertainties through $e^x$, so $e^{x \pm dx}$. I've seen several things online, googling, that are contradictory or unclear, and none of which provide a realistic answer; that is, in most cases the actual uncertainty becomes larger than $x$. I've tried $e^{x + dx} - e^x$, I've tried $\dfrac{e^{x + dx} - e^{x - dx}}{2}$, and neither of these actually work.

What is the proper way to propagate an uncertainty through $e^x$?

Second question concerns a rather complex equation containing several variables with uncertainties. I understand that usually, the derivative of the equation is used...but this isn't the kind of equation you really want to take the derivative of. I'm not even sure I know how, given that the variable I'm trying to calculate is in the equation itself:

$$T_{C} = \frac{\frac{E_A}{R}}{ln \left ( \frac{ART_{C}^2 \frac{D_{0}}{r^2}}{E_{A} \frac{dT}{dt}} \right )}$$

where the variables are A, R, Ea, Tc, D0/r2 (one variable), and dT/dt (one variable); variables with uncertainties are Tc, Ea, and D0/r2, and A, R, and dT/dt are constants.

So I've come up with the following equation, which tries to keep the constants A, R, and dT/dt in place while accounting for the fractional uncertainties of Tc, Ea, and D0/r2 by adding them in quadrature. Brace yourself, this equation is big and brutal (also, I've separated constants A, R, and dT/dt so what's in the natural log term is arranged slightly differently):

[; \delta T_C = \sqrt{(\frac{E_A}{}R)(\frac{\delta E_A}{E_A})^2 + R/\left [ ln\left ( (\frac{AR}{\frac{dT}{dt}})\frac{T_C^2 (D_0/r^2)}{E_A} + (\frac{AR}{\frac{dT}{dt}})\frac{T_C^2 (D_0/r^2)}{E_A}\sqrt{2(\frac{\delta T_C}{T_C})^2 + (\frac{\delta E_A}{E_A})^2 + (\frac{\delta D_0/r^2}{D_0/r^2})^2} \right ) - ln \left ( (\frac{AR}{\frac{dT}{dt}})\frac{T_C^2 (D_0/r^2)}{E_A} \right )\right ]^2} ;]

Basically, the first term, Ea/R is simple, and within the natural log I simply add the term to itself multiplied by the fractional uncertainties in quadrature, minus the original term inside the natural log function. Added in quadrature to the uncertainty of the first Ea/R term. Am I on the right track? Is there an easier way to do this? What would you do if you found yourself trying to propagate the uncertainties through this equation? Any help is appreciated. Thanks much.

• This website uses MathJax. – Silvia Ghinassi Apr 5 '16 at 20:49
• Hey y'all, I answered the first question. The solution is to use the power series expansion of e^x, such that [; \delta e^x = \delta x + \frac{2 \delta x}{2!} + \frac{3 \delta x}{3!} + \frac{4 \delta x}{4!} ... ;] – Adam G Apr 5 '16 at 21:33