5.2 Finding The Derivative
Obviously there need to be simple ways of finding the derivative when it exists. There are
rules of derivatives which make finding the derivative very easy. In the following theorem,
the derivative could refer to right or left derivatives as well as regular derivatives.
Theorem 5.2.1 Let a,b be numbers and suppose f′
exist. Then the
following formulas are obtained.
The formula, 5.2 is referred to as the product rule.
exists and g′
also exists and
This is called the chain rule. In this rule, for the sake of simiplicity, assume the
derivatives are real derivatives, not derivatives from the right or the left. If f
where n is any integer, then
Also, whenever f′
where this definition can be adjusted in the case where the derivative is a right or left
derivative by letting h > 0 or h < 0 only and considering a one sided limit. This is
with the limit being one sided in the case of a left or right derivative.
Proof:5.1 is left for you. Consider 5.2
Since f is differentiable at t it follows that f is continuous there by Theorem 5.1.4. Then
by Theorem 4.11.6 limh→0f
and so, the limit of the right side exists as
0 and equals
This shows 5.2.
Next consider the chain rule. Let
. This follows from continuity of
= 0 and so if this difference is not equal to 0, it will be very small,
is small enough, and the difference quotient is close to f′
case is that g
In this case, H
which is as close as it is
possible to be to
The last claim follows from Example 5.1.2 in case n is a positive integer. If n is 0,
then the claim is obvious. It remains to consider the case where n is a negative integer.
First consider f
For all t≠0, the limit of this last expression is −
using the properties of the limit.
and so, by the chain rule and what was just shown for positive exponent n,
showing that the claim holds in this case also.
Corollary 5.2.2 Let f′
both exist and g
0, then the quotient rule
Proof: This is left to you. Use the chain rule and the product rule. ■
Higher order derivatives are defined in the obvious way.
etc. Also the Leibniz notation is defined by
and the second derivative is denoted as
with various other higher order derivatives defined similarly.
The chain rule has a particularly attractive form in Leibniz’s notation. Suppose
y = g
= g ∘ f
Then from the above theorem
or in other words,
Notice how the du cancels. This particular form is a very useful crutch and is used
extensively in applications.