Math Guide

Partial Differentiation I

Differentiationis a process of finding the derivatives. That is, to find theinstantaneous rate of change of functions at any point. Why we needto find the instantaneous rate of change? It is because the rate ofchange of most of the functions varies from time to time. Forexample, you are driving a car.


The speed at which you are driving isthe rate of change of distance with time. It is never constant andwhat the odometer shows is the speed at that instant, in other wordsthe instantaneous rate of change, In calculus terms that is thederivative of the function of distance with time.


Inmany cases a function depends on more than one variable. That is, thefunction changes with change in any or all the parameters thatcontrol the function. For example, the volume of a cylinder iscontrolled by two factors, the radius and the height. Change ineither or both of these factors changes the volume. 


Therefore, in case of multivariable functions, the derivatives have to be studied with respect to each variable. The process of finding the derivative of the function with respect to a particular variable is called partial differentiation. It may be noted that when partially differentiating a function with respect to a particular variable, all other variables are considered as constants.


Suppose z is a function of only x,then the complete derivative is denoted as dz/dx. On the other hand if z isa function of and as well, then the partial derivative with respect to xis denoted as dz/dx and with respect to yis denoted as dz/dx.


Asone of the partial differentiation examples, let us consider the samecase of volume of a cylinder. The formula is V = πr2h.The instantaneous rate of change of the volume for a change in radiusis expressed as dV/dr,which is 2πrh, treating has constant. The same for a change in height is expressed as dV/dh,which is πr2,treating ras constant.


Therules of normal differentiation are equally applicable to partialdifferentiation. In particular let us highlight the partialdifferentiation chain rule with the same example. Suppose the volumeof a cylinder varies continuously with time due to continuousvariation of both height and radius, then,                          dV/dt= dV/dr*dr/dt+ dV/dh*dh/dt.


Partialdifferential equations are set up as per the studies of partialderivatives. As one of the partial differential equations examples,we can show the heat transfer rate in a particular direction. It isgiven as,du/dt= k(du/dt2),where ‘k’ is a constant.