Editing Total derivative (section)

==Application to equation systems==

In [[economics]], it is common for the total derivative to arise in the context of a system of equations.<ref name=Chiang/>{{rp|pp. 217–220}} For example, a simple [[supply and demand|supply-demand system]] might specify the quantity ''q'' of a product demanded as a function ''D'' of its price ''p'' and consumers' income ''I'', the latter being an [[exogenous variable]], and might specify the quantity supplied by producers as a function ''S'' of its price and two exogenous resource cost variables ''r'' and ''w''. The resulting system of equations
:<math>q=D(p, I),</math>
:<math>q=S(p, r, w),</math>
determines the market equilibrium values of the variables ''p'' and ''q''.  The total derivative <math>dp/dr</math> of ''p'' with respect to ''r'', for example, gives the sign and magnitude of the reaction of the market price to the exogenous variable ''r''.  In the indicated system, there are a total of six possible total derivatives, also known in this context as [[comparative statics|comparative static derivatives]]: {{math|''dp'' / ''dr''}}, {{math|''dp'' / ''dw''}}, {{math|''dp'' / ''dI''}}, {{math|''dq'' / ''dr''}}, {{math|''dq'' / ''dw''}}, and {{math|''dq'' / ''dI''}}.  The total derivatives are found by totally differentiating the system of equations, dividing through by, say {{math|''dr''}}, treating {{math|''dq'' / ''dr''}} and {{math|''dp'' / ''dr''}} as the unknowns, setting {{math|1=''dI'' = ''dw'' = 0}}, and solving the two totally differentiated equations simultaneously, typically by using [[Cramer's rule]].