MthT 430 Notes Chapter 10 Differentiation Best for printing: chap10.pdf
MthT 430 Notes Chapter 10 Differentiation
The purpose of learning differentiation technique is to develop formulas for almost any function defined by a formula.
From the Chapter 5, Limit Theorems, we easily obtain for sums (differences) of differentiable functions:

Theorem 3. If f and g are differentiable at a, then f ±g is differentiable at a, and
(f ±g)′(a)
= f′(a) ±g(a).
More briefly,

Theorem 3. If f and g are differentiable [on a set I], then f ±g is differentiable [on I], and
(f ±g)
= f′±g′.
Product Rule for Differentiation

Theorem 4. If f and g are differentiable at a, then
(f ·g)(a)
= f′(a) ·g(a) + f(a) ·g′(a).
Proof.p. 167.
The idea behind the proof: Let ∆h f = f(a +h) − f(a), etc., so that

f′(a) =
lim
h → 0 
h f

h
,
etc.,
(f ·g)(a + h)
= (f(a) + ∆h f) ·(g(a) + ∆h g)
= f(a)·g(a) + ∆h f ·g(a) + f(a) · ∆h g + ∆h f ·∆h g,
h (f·g)
= (f ·g)(a + h) − f(a)·g(a)
= ∆h f ·g(a) + f(a) · ∆h g + ∆h f ·∆h g,

lim
h → 0 
h (f·g)

h
=
lim
h → 0 
h f

h
·g(a) + f(a) ·
lim
h → 0 
h g

h
+
lim
h → 0 
h f

h
·
lim
h → 0 
h g
= f′(a) ·g(a) + f(a) ·g′(a).
The derivative of the nth power function is given by:

Theorem 6. If f(x) = xn for some natural number n, then
f(a)
= n an−1
(♠)
for all a.
Derivative of a Quotient

Theorem 7. If g is differentiable at a, and g(a) ≠ 0, then 1/g is differentiable at a, and
(1/g)(a)
= − g′(a)

(g(a))2
.
Proof. p. 167. An alternate proof is to observe that 1/g is the composition of the oneover function with g and use the chain rule .

Theorem 8 (Quotient Rule). If f and g is differentiable at a, and g(a) ≠ 0, then f/g is differentiable at a, and

f

g



 
(a)
= g(a)·f(a) − f(a) ·g(a)

|g(a)|2
.
An alternate form:

u

v



 
= v u − v u

v2
.
From an old calculus book (Robert Bonic, et al., Freshman Calculus):

u

v



 
:                                                           

v2
(write vinculum over v2)
v                                                        

v2
(write v again before you forget)
v u − v u

v2
(fill in the rest)
Some prefer the forms:

u

v



 
=

u ·
1

v





 
= u′

v
− u ·
v′

v2

,

(u v−1)′
= u′v−1 + (−1) u v v−2 .
(♣)
The form (♣) is particularly convenient for repeated derivatives.
Chain Rule

Theorem 9 (Chain Rule). If g is differentiable at a, and f is differentiable at g(a), then f°g is differentiable at a and
(f °g)(a)
= f(g(a)) ·g(a).
Other forms of the Chain Rule
Leibniz notation:
y
= y(u),
dy

dx
= dy

du
  du

dx
.
Leibniz (mixed) notation:
df(u(x))

dx
= f(u(x))· du

dx
.
Function notation:
(f °g)
= (f′°g)·g′
One layer at a time:
df(♥(…))

dx
: f(                                                        )
(deriv of outside function f)
→ f(♥(…))
(evaluate at inside function)
→ f(♥(…))·
(TIMES)
→ f(♥(…))·
d(♥(…))

dx

(deriv of inside)
(do it again if necessary)



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On 05 May 2013, 19:44.