Page 27 - Essentials of physical chemistry

Introduction: Mathematics and Physics Review xxv

n n (n 1) n 2
(x þ h) ¼ x þ nhx þ þ h so that all the terms of h or higher will go to zero and in the
derivative of any variable to an integer power n, THE n FALLS OFF THE ROOF, LEAVING
(n 1) as the exponent! That is all we need to know for derivatives of simple polynomials! So we
n
have a simple but powerfully general formula: given y ¼ x , we can immediately write an easily
remembered formula as

n
dx (n 1)
¼ nx :
dx
This may seem a very informal way to remember this process but in the interest of mental efficiency,
it will get the job done and we will seek other correct shortcuts for mathematical operations. (Life is
short and we have many equations yet to learn!)
The next special derivative we will use many times involves the derivatives of exponential
functions of base ‘‘e.’’ The number ‘‘e’’ was implied by the mathematician Napier in 1618,
developed further by Bernoulli, and called ‘‘e’’ by Euler in 1727. It is an irrational number like
p, but for our purposes we can use a good approximation on our calculator as

e ffi 2:7 1828 1828 45 90 45 23536 ...

For those interested in science facts, e can be remembered as roughly 2.7 followed by the digits
1
1828, 1828 again, and then 45, 90, and 45 but usually you can just enter e on your calculator to
verify the approximation. Bernoulli attempted to evaluate the constant using the formula
n
1
e ¼ lim 1 þ ,
n
n!1
and you can easily check the first few digits with your calculator. Napier determined that only one
base leads to an exponential function whose functional graph has the property that ALL the
derivatives of that function (especially the first derivative) have the same value at any point on
the graph as the value of the function itself!

x x x n
de x d de x d d de x d x x
¼ e , ¼ e , ¼ e , and e ¼ e for any n:
dx dx dx dx dx dx dx
That in itself is merely a profound fact but most importantly for us is the similarity to the polynomial
formula above when the exponent ‘‘falls off the roof.’’
d ax ax
(e ) ¼ ae :
dx
The proof of this derivative is slightly complicated but for our purposes we can memorize it. It is
instructive here to introduce the concept of the CHAIN RULE for derivatives of functions, which
u
ax
are in turn functions of variables as for instance when u ¼ ax in e ¼ e . Then we have to take the
derivative with respect to u and the chain derivative of u with respect to x so we have:
u
de u de ax de du ax
¼ ae :
¼
dx dx du dx
¼

d du d
Note that the use of the chain rule leads to the ratio ¼ .
du dx dx
Now consider the case where we can introduce another formula that you may or may not have
seen in one semester of calculus for the Taylor expansion of any function of (x) in terms of all its
x
derivatives as applied in this case to e . It is worth mentioning the Taylor expansion in this