Till the previous post, we studied the concept of differentiation w.r.t just one variable. However, quite often, we encounter situations which involve more than one variable. The concept of ‘Partial derivative’ comes into the picture in such situations.
PARTIAL DIFFERENTIATION
Consider a function y = f(x). In this equation, y is the only variable.
The derivative, f'(x) = dy/dx ⇒ represents the rate of change of ‘y’ as ‘x’ changes. We studied how to differentiate such functions earlier.
However, in science, we deal with many parameters at once. We encounter entities that are dependent on more than one quantity.
e.g.– In thermodynamics, Pressure(P), Volume(V), Temperature(T), Internal Energy (E), Entropy(S) define a thermodynamic system. Most of the time, more than one of these quantities varies and we have to deal with more than one variable, at one time. In such situations, we use the concept of partial differentiation.
A partial derivative of a function (of several variables), is its derivative with respect to one of those variables, with the others held constant.
Consider a function, u = f (x,y) ☞ this means ‘u’ is dependent on both ‘x’ & ‘y’. Then,
the partial derivative of the variable u w.r.t x is (∂u/∂x)y → y is kept constant.
the partial derivative of the variable u w.r.t y is (∂u/∂y)x → x is kept constant.
Then by fundamental partial differentiation theorem,
Total differential of u = du= (∂u/∂x)y dx + (∂u/∂y)x dy.
The symbol ‘∂‘ (called ‘del‘) is used to represent a partial derivative. ‘d’ is used to represent full derivatives.
We shall come across many examples of partial derivatives in thermodynamic, quantum mechanics, and spectroscopic studies.
I end the discussion on derivatives with this post. I hope understanding the basics will help us learn our chemistry chapters in a better way. In the upcoming post, we shall begin our discussion on the second part of calculus- INTEGRATION. Till then,
Be a perpetual student of life and keep learning…
Good Day!
References and Further Reading –
1)https://www.tcd.ie/Physics/cpam/teaching/thermodynamics/notes/Thermomath.pdf
2)http://www.columbia.edu/itc/sipa/math/calc_rules_multivar.html
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