Differentiation

Recall the basic definition for a derivative

This definition gives the instantaneous slope of a line tangent to a curve.

We also write the derivative as f’(x) [f prime of x]. From the above equation with just a little algebra you can derive the general formula for polynomials

There are a few basic rules that will allow you to apply this to a large number of functions.

The product rule states that

The quotient rule states that

If you have a hard time remembering the order of f(x) and g(x) in the quotient rule you can also treat f(x)/g(x) as the product of f(x) and 1/g(x). This has the form

Which is completely equivalent to the quotient rule. Note that we used the polynomial rule here since 1/g(x) = g(x)^-1. In general, if you are given a function in the denominator just write as a negative exponent first. This will make taking the derivative much easier.

Example,

where we treated the function 1/f(x) as f(x)^-1 and therefore n = -1 and nf(x)^n-1= (-1)f(x)^-2. In this example, we have used the chain rule. The chain rule applies when one function is “buried” inside another, e.g. g(f(x)).

First, take the derivative with respect to g(x) treating the whole of f(x) as the variable, then take the derivative with respect to f(x).

Example,

In this example, we take the derivative of a Gaussian function with respect to x. Note that here g(f(x)) = e^f(x) and f(x) = -ax². The derivative of an exponential is the exponential itself times the derivative of the exponent.

Functions of more than one variable

If we have a function of more than one variable, f(x,y) we can take the derivative with respect to either one. These are called partial derivatives with respect to x or y (or whatever the variable is).

The partial derivative with respect to x is

The partial derivative with respect to y is

The total derivative is

We say the total derivative is an exact differential is the second cross derivatives are equal

If these cross derivatives are not equal the total derivative is not an exact differential.

In physical chemistry this is important becuase

State functions are exact differentials

Path functions are inexact differentials

A state function has the same magnitude regardless of the path taken.

The integral has the same magnitude regardless of the path taken if the total derivative of x is exact.

If the total derivative is not exact then

For example, in thermodynamics we show that the internal energy is a state function, but the work and the heat are path functions.