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 7.5.1Let a,b be numbers and suppose f^{′}
(t)
and g^{′}
(t)
exist. Then thefollowing formulas are obtained.
(af + bg)′(t) = af′(t) + bg′(t). (7.14)
(7.14)
′ ′ ′
(fg)(t) = f (t)g (t)+ f (t)g (t) . (7.15)
(7.15)
The formula, 7.15is referred to as the product rule.
If f^{′}
(g(t))
exists and g^{′}
(t)
exists, then
(f ∘g)
^{′}
(t)
also exists and
′ ′ ′
(f ∘g) (t) = f (g(t))g(t).
This is called the chain rule. In this rule, for the sake of simiplicity, assume the derivativesare real derivatives, not derivatives from the right or the left.If f
(t)
= t^{n}where n is anyinteger, then
f′(t) = ntn− 1. (7.16)
(7.16)
Also, whenever f^{′}
(t)
exists,
f′(t) = lim f (t+-h)−-f (t)
h→0 h
where this definition can be adjusted in the case where the derivative is a right or leftderivative by letting h > 0 or h < 0 only and considering a one sided limit.This is equivalentto
′ f (s)−-f-(t)
f (t) = lis→mt t− s
with the limit being one sided in the case of a left or right derivative.
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.
′ ( f (t+-h)−-f-(t) o(h)) f (t+-h)−-f (t)
f (t) = lhim→0 h + h = lhim→0 h .■
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
(h)
does.
Corollary 7.5.2Let f^{′}
(t)
,g^{′}
(t)
bothexist and g
(t)
≠0, then the quotient ruleholds.
( )
f-′ f′(t)g(t)−-f (t)g′(t)
g = g(t)2
Proof: This is left to you. Use the chain rule and the product rule. ■
Higher order derivatives are defined in the usual way.
f′′ ≡ (f′)′
etc. Also the Leibniz notation is defined by
dy-= f′(x) where y = f (x)
dx
and the second derivative is denoted as
2
d-y
dx2
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
(u )
and u = f
(x)
. Thus y = g ∘ f
(x )
. Then from the above theorem
(g∘f)
^{′}
(x)
= g^{′}
(f (x))
f^{′}
(x)
= g^{′}
(u)
f^{′}
(x)
or in other words,
dy- dydu-
dx = dudx .
Notice how the du cancels. This particular form is a very useful crutch and is used extensively
in applications.