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Chain Rule
Objectives
Students will be able to
• Apply the chain rule to find derivatives.
Rules
Product Rule
• If g is differentiable at x0 and f is
differentiable at u0 = g(x0) then F (x )  f ( g (x ))
is differentiable at x0 and the derivative is
F (x 0 )  f (u0 ) g (x 0 )  f ( g (x 0 )) g (x 0 )
Example 1
Let f (x )
 5x  2x and g (x )  8x  3 . Find
2
f[g(2)] and g[f(2)]
Example 2
2
Let f (x )  4 and
x
f[g(x)] and g[f(x)]
g (x )  2  x . Find
Example 3
Find the derivative of
y  2x  9x 
3
5
Example 4
Find the derivative of
y  3 7t  1
3
Example 5
Find the derivative of
x  4x
p (t ) 
(3x 3  2) 4
2
Example 6-1
Suppose the cost in dollars of manufacturing q
items is given by
C  2000q  3500
and the demand equation is given by
q  15000  1.5p
In terms of the demand
a. Find an expression for the revenue R.
Example 6-2
Suppose the cost in dollars of manufacturing q
items is given by
C  2000q  3500
and the demand equation is given by
q  15000  1.5p
In terms of the demand
b. Find an expression for the profit P.
Example 6-3
Suppose the cost in dollars of manufacturing q
items is given by
C  2000q  3500
and the demand equation is given by
q  15000  1.5p
In terms of the demand
c. Find an expression for the marginal profit.
Example 6-4
Suppose the cost in dollars of manufacturing q
items is given by
C  2000q  3500
and the demand equation is given by
q  15000  1.5p
In terms of the demand
c. Determine the value of marginal profit when
the price is $5000.