Section 4.6: Derivatives of Exponential Functions
Theorem: (Derivative of the Exponential Function)
If f (x) = ex , then f 0 (x) = ex . By the Chain Rule, it follows that
d f (x)
[e ] = f 0 (x)ef (x) .
dx
Example: Differentiate each function.
(a) f (x) = e4x
2 −2x+1
2
(b) f (x) = e1−3x cos(5x)
(c) f (x) = ex tan x
1
(d) f (x) =
1 + x2
2 + e−x
(e) f (x) = sin
1 − e2x
1 + e2x
2
Example: Show that if f (x) = ax , then f 0 (x) = (ln a)ax .
Theorem: (Derivative of a General Exponential Function)
d x
(a ) = (ln a)ax
dx
By the Chain Rule, it follows that
d f (x)
[a ] = (ln a)af (x) f 0 (x).
dx
Example: Differentiate each function.
(a) f (x) = 2x
3 −1
√
(b) f (x) = 4
1−2x3
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Example: (Bacterial Growth) Suppose that the population size of a bacterial colony at time
t ≥ 0 is given by
N (t) = N0 ert ,
where N0 is the initial population size and r > 0 is the growth rate. Show that
dN
= rN.
dt
What is the per capita growth rate?
Example: (Radioactive Decay) Suppose that y(t) denotes the mass of a radioactive substance
left after t days. If the decay rate of the material is 0.3/day, find a differential equation for
the decay function y(t).
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Example: (Radioactive Decay) Suppose that y(t) denotes the mass of a radioactive substance
left after t days. Assume that y(0) = 15 and
dy
= −3y.
dt
(a) Find an expression for the mass left at time t.
(b) How much material is left after t = 3 days?
(c) What is the half-life of this material?
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Example: (Von Bertalanffy Growth Model) The growth of fish can be described by the von
Bertalanffy growth function
L(x) = L∞ − (L∞ − L0 )e−kx ,
where x denotes the age of the fish and k, L∞ , and L0 are positive constants.
(a) Evaluate L(0) and interpret the biological meaning of L0 .
(b) Evaluate lim L(x) and interpret the biological meaning of L∞ .
x→∞
(c) Differentiate L(x) and show that the growth rate satisfies the differential equation
dL
= k(L∞ − L).
dx
What does this proportionality say about how the growth rate changes with age?
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