What exactly is a partial derivative? I have a basic idea of a derivative i.e, I know that dy/dx is rate of change of y with respect to x and that it represents the slope of a curve
Thanks!
In: Mathematics
When explaining things I like to use a practical example. The ideal gas law is written as PV=nRT and we will rewrite it as P=nRT/V so we now have a function where Pressure (P) is expressed as an equation related to three variables the number of moles of a gas (n) the temperature of a gas (T) and the volume of the container (V). Now if n and V are constant than the equation is just expressed as a function P(T) and the derivative P'(T) can be expressed as dP/dT but if there is more than one variable we cannot take a single derivative.
Remember that the derivative is nothing more, and nothing less, than the rate at which the equation is changing. For a multi-variable equation that equation must be broken down into components expressed as a vector. These components are the partial derivatives. Consider again our example P=(nRT)/v since it is a function P(n,T,V) it would be expressed as three partial functions added together. So P'(n,T,V)=pP/pn+pP/pT+pP/pV (actually signified by a lowercase delta but I don’t have that on my phone’s keyboard). Which is found by essentially pretending the other variables are constant.
So pP/pn is (RT/V)pn, pP/pT is (nR/V)pT, and pP/pV is (-nRT/V^2)pV.
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