4.1 – Simple Derivative Rules and Applications
A. - Simple Derivative Rules
function
derivative formula
I. Constant
y = f (x) = constant
y = f ( x) = 0
(Recall: derivative is slope and slope of a horizontal line (y = const) is 0.)
(Note: rule of thumb for alphabet: usually the beginning of the alphabet denotes
constants and the end of the alphabet denotes variables.)
II. Power Rule
y = f ( x) = xn , (n  )
III.
Constant Times a Function Rule
y = f ( x) = k  g( x)
IV.
y = f ( x) = n x n 1
y = f ' ( x) = k [ g ' ( x)]
Sum and Difference Rule
y = f ( x) = g ( x) ± h ( x)
y = f ( x) = g( x) ± h( x)
V. Exponential and Log
(a)
(b)
y = ex
y = ln x , x > 0
y = f ( x) = e x
y = f ( x) =
1
x
B. - Marginal Analysis - analyzing cost, revenue and profit function derivatives
Recall:
Total Cost = (variable costs) (quantity produced) + (fixed costs)
Total Revenue = (price per unit) (quantity sold)
Total Profit = Total Revenue - Total Cost
Marginal Cost, Revenue, and Profit
Marginal Cost =
one more item)
rate of change of total cost
=
derivative of total cost (the approximate cost of
Marginal Revenue = rate of change of total revenue
approximate income from one more item)
Marginal Profit
one more item)
= rate of change of total profit
=
derivative of total revenue (the
= derivative of total profit (the approximate profit from
4.2 - Derivatives of Products and Quotients (and applications)
A. - Derivative Rules
I.
Product Rule
y  f (x) = g( x)  h( x)
y = f (x) = g(x)  h(x) + h(x)  g(x)
OR
y = f ( x) = g( x)  h( x) + h( x)  g( x)
(since multiplication is commutative)
II.
Quotient Rule
y = f ( x) =
t ( x)
b( x)
y = f ( x) =
b( x)  t ( x)  t ( x)  b( x)
b(x)2
(don’t forget that subtraction is not commutative,
so you cannot switch the order in the numerator)
B. - Marginal Average Cost, Revenue, and Profit
If C(x) is the cost of producing x items, then the average cost C (x) is the cost of all the items
divided by the number of items: C ( x) 
C( x)
.
x
Then Marginal Average Cost = C ( x) 
d
d C(x)
and
C (x) 
dx
dx x
R (x) 
d
d R( x)
and
R ( x) 
dx
dx x
Marginal Average Revenue =
Marginal Average Profit = P ( x) 
d
d P( x)
P ( x) 
dx
dx x
4.3 - The (all important) Chain Rule
A function h is a composite of functions f and g if h( x)  f [ g(x)].
Then
d
f [ g(x)]  f [ g(x)] g(x)
dx
General Power Rule:
function
(a)
derivative
y  f (x) =  u( x)  n
y = f ( x) = n   u (x)  n 1  u( x)
(Notice that we now have a function raised
to a power and not just x raised to a power)
EXAMPLES:
1. Find the derivative for the following functions and simplify:
(a)
f ( x)  x  2 
3
x2  4 2
(b)
f (x)  ex  2ln x
(c)
e x  x2
f (x) 
6
(d)
x
f (x)  (2x)5  e x  2  3ln 
 4
(e)
f ( x)  e4  ln x
2. Find the derivative for the following functions and simplify:
(a)
(b)
f ( x)  5x2 ( x3  2)
f ( x)  x3 h( x)
(c)
f ( x)  (5  x2 )(x3  2x2 )
(d)
f (x) 
h( x)
x2
(e)
f ( x) 
63 x
x2  3
3. Find the equation of the line tangent to the graph of f ( x)  (7  3x )(x  2x 2 ) at x = 2.
4. Find the value(s) of x where f ( x) 
x
has horizontal tangent lines.
x2  1
5. The price equation for the production of television sets is given by x  9,000  30 p and the cost
equation is C(x)  150,000  30x , where x is the number of sets that can be sold at a price $p
per set and C(x) is the total cost (in dollars) of producing x sets.
(a) Find the Revenue and Profit functions.
(b) Find the marginal cost, revenue and profit functions.
(c) Evaluate marginal cost, revenue and profit at x = 2000 and interpret.
(d) Find the exact profit of the 200th set. Use marginal profit to estimate the profit of the 200th set.
(e) Sketch Cost and Revenue together and interpret.
6. A machine shop makes drill bits with C( x)  1000  25x  0.1x 2 .
a. Find the average cost function for drill bits.
b. Find the average cost of producing a drill bit at a production level of 10 drill bits per day and
interpret.
c. Estimate the average cost per bit at a production level of 11 drill bits per day and interpret.
7. Find the derivative for the following functions and simplify:
(a)
f ( x)  x 2 2x3  1
(b)
( x  5)5
f ( x) 
(1  3x) 4
(c)
f ( x)  ( x2  1)4 (6  2x)3
(d)
ex
f ( x)  2
( x  4)2