Appendix  Propagation of Errors ( Calculus Method )
PROPAGATION OF ERRORS
When taking measurements in the laboratory, there is an uncertainty associated with each
measurement.
All of the measurements are then combined in some way to give a final
experimental value.
Since there is an uncertainty in the individual measurements, there has to be
an uncertainty in the final answer.
The correct way to find this associated uncertainty of the final
answer is through the method of propagation of errors.
In this method, which
is described
below, you will see how to take into account these individual measurement errors and how they
are propagated through to the final answer.
This technique requires the use and knowledge of partial derivatives.
Since you may or
may not know how to take partial derivatives, we will teach you the method, but not the theory.
The theory will be taught in Calculus III and you do not need to be exposed to it to understand
the techniques of taking partial derivatives.
The method of taking partial derivatives is very
similar to taking the derivative of a function with a single variable.
The best way to learn how to do partial derivatives is by example. The important thing to
notice here is that when taking the partial derivative of a function of several variables with
respect to one of those variables, you treat all of the other variables as constants
.
This should
hopefully become clear in the following example.
Example 1 
Let's look at the function
f(x,y,z) = 2xy
2
z
3
+ {x
3
/z
2
} = 2xy
2
z
3
+ x
3
z
2
(14.1)
As you can see, the function depends on three variables.
This means that there are three partial
derivatives that you can take of this function
f
.
The proper way to write the partial derivative of
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 Spring '08
 ELLIS
 Derivative, TA, Sausalito

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