4.3 - 4.5: Chain Rule, Rules for Exponential and Logarithmic Functions, and Elasticity
The Chain Rule: Given y = f (g(x)). If the derivatives g 0 (x) and f 0 (g(x)) both exist, then y 0 exists
and
(f (g(x)))0 = f 0 (g(x)) · g 0 (x).
Using Lebniz notation, if y = f (u), u = g(x) and they are differentiable functions, then
dy du
dy
=
·
.
dx
du dx
Example 1. Find the derivatives of the following functions:
a) F (x) = (−2x3 + x2 − 3x + 7)6 .
b) G(x) = (9x2 + 6x + 1)9 (2x + 4)5 .
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c) H(x) = (5x2 − 6x + 7)−11 .
d) K(x) =
√
x3 − 3x2 + 3x − 1.
e) L(x) =
1
.
6x + 3
f ) f (t) =
2 + t2
t
− 27
.
5
2z + 3
2
.
g) g(z) = z +
1 − 2z
2
3
h) h(w) = p
.
3
−ln(w) + 2
−2
y−1
i) k(y) = 4 +
.
y+1
2
j) F (s) = e3s .
k) G(t) = (ln(2t + 1))5 .
l) y = ln(x3 ).
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m) h(r) =
(r3 + 4)−3
.
r2 + r + 1
n) v(x) =
(10x7 + 5x5 )2
.
x+1
o) u(t) =
r
4
3
(2t + 1) + 5 .
p) f (x) = |x2 + 5x|.
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Example 2. Find the equation of tangent line at the point with x = −1 of the graph of
√
f (x) = x3 + 2ln(e2x ).
Example 3. Given H(x) = f (g(x)), and g(−2) = 7, g 0 (−2) = 4, f 0 (−2) = 5, and f 0 (7) = 1. Find
H 0 (−2).
Example 4. Given g(x) =
√
xlnx. Find the derivative of g(x) at the x making f (x) = 2e.
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More Derivative Rules:
• The Exponential Rule:
d x
a = (lna)(ax ).
dx
• The Logarithmic Rule:
d
1
(loga x) =
.
dx
xlna
Note: Derivative of Logarithmic Function:
d
f 0 (x)
(ln|f (x)|) =
.
dx
f (x)
Example 5. Differentiate:
x2
2
a) f (x) =
.
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b) g(x) = log5 (5x5 ).
√
c) y = x2 ln| x − 7|.
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Example 6. Given
f (x) =
lnx
,
ln(x3 − x2 )
find f 0 (2e).
Example 7. Find the equation of tangent line at (1,25) of the graph of
y = 52x .
Example 8. The equation
100
1 + 50e−0.05t
is used to model the population of an area in millions, where t is the calendar year. Find the rate of
change of this population with respect to time. What is the sign of this rate?
p(t) =
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Example 9. The count in a bacteria culture was 500 after 2 hours and 50,000 after 8 hours.
a) What was the initial population of the culture? (Round to the nearest bacterium.)
b) Find the population after t hours.
c) How fast is the population growing after 10 hours? (Round to 4 decimal places.)
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Example 10. Polonium-210 has a half-life of 140 days. (Round your answers to 4 decimal places.)
a) If a sample has an original mass of 300 mg, find the amount of mass remaining after t days.
b) How long does it take the mass to reduce to 20 mg?
c) How fast is the mass decaying when 20 mg remains?
Relative Rate of Change
The relative rate of change of a function f (x) is
f 0 (x)
.
f (x)
The percentage rate of change of a function f (x) is
f 0 (x)
100 .
f (x)
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Example 11. Find the relative rate of change of f (x) = 5x + ln|x2 − 2x|.
Example 12. Stock P costs $20 per share. Stock Q costs $80 per share. If it can be estimated that
Stock P’s earning will increase $2 per year over the next few years and Stock Q’s earning will increase
$5 per year over the next few years, then which stock is a better buy?
Elasticity of Demand
Elasticity is a measure of sensitivity of demand to price change of a product.
• Elastic demand: Demand is sensitive to changes of price, i.e. the percentage change in demand
is greater than the percentage change in price.
• Inelastic demand: Demand is not sensitive to changes of price, i.e. the percentage change in
demand is less than the percentage change in price.
Point elasticity is used to find the elasticity at a single quantity, provided the demand is a function of
price of the form x = d(p).
Example 13. Find the demand equation x = d(p) for the price-demand function
p=
500x
.
3x + 4
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The elasticity of demand, denoted by E(p), of a product at a price p is the negative ratio of the
relative rate of change in demand to the relative rate of change in price:
dx
p dx
p · d0 (p)
=−
.
E(p) = − x = − ·
dp
x dp
d(p)
p
• If E(p) > 1, then the demand is elastic, i.e. relatively large changes in demand.
• If E(p) < 1, then the demand is inelastic, i.e. relatively small changes in demand.
• If E(p) = 1, then the demand has unit elasticity, i.e. relatively equal changes in demand.
For problem solving:
E(p) =
%change in quantity
.
%change in price
Example 14. Given the price-demand equation
p + 0.01x = 50.
a) Find the demand x as a function of p.
b) Find the elasticity of demand E(p).
c) What is the elasticity of demand when p = $15? If this price is decreased by 10%, what is the
approximate change in demand?
d) What is the elasticity of demand when p = $25? If this price is decreased by 10%, what is the
approximate change in demand?
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e) What is the elasticity of demand when p = $45? If this price is decreased by 10%, what is the
approximate change in demand?
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Example 15. Assume that the price of a product is $100 and E(100) = 0.5. If the price is increased
by $10, then what is the approximate change in demand? (Round to the neared hundreth of a percent).
Remark: If the demand is elastic, then a change of price will lead to a larger change in the quantity
demanded. Hence, if E > 1, increasing price decreases revenue. If the demand is inelastic, then a
change of price will lead to a smaller change in the quantity demanded. Hence, if E < 1, increasing
price increases revenue. Revenue is optimized when E = 1.
√
Example 16. Given the demand function, d(p) = 150 − 3p, determine the elasticity of demand at
the following points. Classify the demand as elastic, inelastic or unit elastic at each point. Determine
whether the price of the item should be raised or lowered from each point in order to increase revenue.
a) p = 30.
b) p = 50.
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Example 17. Using the price-demand equation x = 10(p − 70)2 , find the values of p for which demand
is elastic and for which demand is inelastic.
Example 18. A company sells hats at $12 each and calculates the demand function for the hats to be
d(p) = 60 − 4p.
a) Is the demand elastic or inelastic at the current $12 price?
b) Should the hat price be raised or lowered to increase revenue?
c) Set E(p) = 1 and solve for p to determine at which price the revenue is greatest.
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