How do we derive the product rule?
2 Answers
Please see below.
Explanation:
Let
Now from definition
also
and
As
=
=
=
=
=
Please see below.
Explanation:
I use the product rule in the form:
(Because multiplication and addition are commutative, it can also be written in other orders.)
Proof:
Suppose that
Let
(We will show that
Then
# = lim_(hrarr0)(f(x+h)g(x+h) - f(x)g(x))/h#
Now we will add
# = lim_(hrarr0)(f(x+h)g(x+h) -f(x)g(x+h)+f(x)g(x+h) - f(x)g(x))/h#
Now regroup:
# = lim_(hrarr0)((f(x+h) -f(x))g(x+h)+f(x)(g(x+h) - g(x)))/h#
Rewrite as the sum of two ratios and factor.
# = lim_(hrarr0)((f(x+h) -f(x))g(x+h))/h + (f(x)(g(x+h) - g(x)))/h#
# = lim_(hrarr0)((f(x+h) -f(x)))/hg(x+h) + lim_(hrarr0)f(x)((g(x+h) - g(x)))/h#
We now have limits of 4 things to evaluate.
# lim_(hrarr0)((f(x+h) -f(x)))/h = f'(x)#
#lim_(hrarr0)g(x+h)=g(x)# (see note (2) below).
#lim_(hrarr0)f(x) = f(x)#
# lim_(hrarr0)((g(x+h) -g(x)))/h = g'(x)#
Finally we write the total limit:
# = f'(x)g(x)+f(x)g'(x)#
That is:
Notes
(1) To prove the product rule in a different order we would add
(2) By hypothesis