Calculus standard derivatives

In this paragraph we will perform a fairly standard calculus and derive the contribution coming from a small loop Wmfl ... 

Solutions to some standard derivatives appear in Table A.3. It does not hurt to know how to obtain derivatives and integrals, but we will be treating these aspects of calculus as just another kind of algebra. In other words, one may replace the derivative or integral expression by the correct algebraic expression, with the appropriate substitutions. This will suffice for almost all the calculus we encounter in the text. 

How is the least square set of coefficients computed The methods of calculus give us the procedure to use it is the standard, first-term calculus procedure take the derivative and set it equal to zero, this gives a minimum. In this case, it is (only slightly) more complicated in that we need to take the derivatives of the error expressions with respect to each bt, since we want to compute the values for all the bt that give the lowest sum-squared error. 

Note that steps 11 through 19 do not require you to use calculus in order to compute the partial derivatives dF/dx, dF/dy, and dF/dz the spreadsheet does that for you, based on (2.4-3). It can always do this as long as it deals with specific numerical values for x, delx, y, dely, z, delz, etc. The above approach illustrates a calculus-free yet perfectly legitimate way to compute the general propagation of standard deviations in any formula, no matter how complicated, provided that the parameters x, y, z, etc. are mutually independent. And it need not add much work because, especially for a complicated formula, you will most likely already use the spreadsheet to calculate the standard deviations anyway. 

Using standard calculus notation, we will use a prime to indicate a derivative with respect to a variable. The previous equation becomes, more succinctly,... 

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