For reactive impedances   Moving here until its right  Jacobi obtained his theorem by logical reasoning but the mathematical proof is as follows  In the diagram opposite power is being transferred from the source with voltage  and fixed source impedance  to a load with resistance  resulting in an instantaneous current    is simply the source voltage divided by the total circuit resistance    The average power  dissipated in the load is the square of the current multiplied by the resistive portion of the load impedance     We could calculate the value of Zload for which this expression is a maximum but it is easier to calculate the value of Zload for which the denominator     is a minimum  The result will be the same in either case  Differentiating with respect to Zload    For a maximum or minimum the first derivative is zero so    or    In practical resistive circuits Zsource and Zload are both positive  To find out whether this solution is a minimum or a maximum we must differentiate again    This is positive for positive values of Zsource and Zload showing that the denominator is a minimum and the power is therefore a maximum when Zsource  Zload