7.5 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 7.5.1 Let a,b be numbers and suppose f′
exist. Then the
following formulas are obtained.
The formula, 7.15 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
tn where n is any
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 equivalent
with the limit being one sided in the case of a left or right derivative.
Proof:7.14 is left for you. Consider 7.15
because by Lemma 7.3.6, g is continuous at t and so
This proves 7.15
Next consider the chain rule. By Lemma 7.3.6 again,
This proves the chain rule.
Now consider the claim about f
an integer. If n
1 the desired
conclusion follows from Lemma 7.4.1
. Suppose the claim is true for n ≥
Then by the product rule, induction and the
validity of the assertion for n
and so the assertion is proved for all n ≥ 0. Consider now n = −1.
Therefore, the assertion is true for n
Now consider f
is a positive
integer. Then f
and so by the chain rule,
This proves 7.16.
Finally, if f′
Divide by h and take the limit as h → 0, either a regular limit or a limit from one side or the
other in the case of a right or left derivative.
Note the last part is the usual definition of the derivative given in beginning calculus
courses. There is nothing wrong with doing it this way from the beginning for a function of
only one variable but it is not the right way to think of the derivative and does not generalize
to the case of functions of many variables where the definition given in terms of o
Corollary 7.5.2 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 usual way.
etc. Also the Leibniz notation is defined by
and the second derivative is denoted as
with various other higher order derivatives defined in the usual way.
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