Page 84 - Calculus Demystified

CHAPTER 2
Rules for Calculating Derivatives
2.5 Foundations of Calculus 71
Calculus is a powerful tool, for much of the physical world that we wish to analyze
is best understood in terms of rates of change. It becomes even more powerful when
we can find some simple rules that enable us to calculate derivatives quickly and
easily. This section is devoted to that topic.

I Derivative of a Sum [The Sum Rule]: We calculate the derivative of a sum
(or difference) by

(f (x) ± g(x)) = f (x) ± g (x).
In our many examples, we have used this fact implicitly. We are now just
enunciating it formally.
II Derivative of a Product [The Product Rule]: We calculate the derivative
of a product by

[f(x) · g(x)] = f (x) · g(x) + f(x) · g (x).
We urge the reader to test this formula on functions that we have worked with
before. It has a surprising form. Note in particular that it is not the case that

[f(x) · g(x)] = f (x) · g (x).
III Derivative of a Quotient [The Quotient Rule]: We calculate the derivative
of a quotient by

f(x) g(x) · f (x) − f(x) · g (x)

= .
2
g(x) g (x)
In fact one can derive this new formula by applying the product formula to
g(x) ·[f (x)/g(x)]. We leave the details for the interested reader.
IV Derivative of a Composition [The Chain Rule]: We calculate the
derivative of a composition by

[f ◦ g(x)] = f (g(x)) · g (x).
To make optimum use of these four new formulas, we need a library of functions
to which to apply them.
k
A Derivatives of Powers of x:If f(x) = x then f (x) = k · x k−1 , where

k ∈{0, 1, 2,...}.
Math Note: If you glance back at the examples we have done, you will notice that
2
we have already calculated that the derivative of x is 1, the derivative of x is 2x,