5.4 Differentiation of Exponential Functions

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5.4 Differentiation of Exponential Functions

• By
• Dr. Julia Arnold and Ms. Karen Overman
• using Tans 5th edition Applied Calculus for the
  managerial , life, and social sciences text

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Lets consider the derivative of the exponential
function. Going back to our limit definition of
the derivative
First rewrite the exponential using exponent
rules.
Next, factor out ex.
Since ex does not contain h, we can move it
outside the limit.
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Substituting h0 in the limit expression results
in the indeterminate form , thus we will
need to determine it.
We can look at the graph of
and observe what happens as x gets close to 0.
We can also create a table of values close to
either side of 0 and see what number we are
closing in on.
Table
Graph
x -.1 -.01 -.001 .001 .01 .1
y .95 .995 .999 1.0005 1.005 1.05
At x 0, f(0) appears to be 1.
As x approaches 0, y approaches 1.
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We can safely say that from the last slide that
Thus
Rule 1 Derivative of the Exponential Function
The derivative of the exponential function is the
exponential function.
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Example 1 Find the derivative of f(x) x2ex .
Solution Do you remember the product rule? You
will need it here.
Product Rule (1st)(derivative of 2nd)
(2nd)(derivative of 1st)
Factor out the common factor xex.
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Example 2 Find the derivative of f(t)
Solution We will need the chain rule for this
one.
Chain Rule (derivative of the outside)(derivative
of the inside)
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Why dont you try one Find the derivative of
.
To find the solution you should use the quotient
rule. Choose from the expressions below which
is the correct use of the quotient rule.
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No thats not the right choice. Remember the
Quotient Rule
(bottom)(derivative of top) (top)(derivative of
bottom) (bottom)²
Try again. Return
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Good work! The quotient rule results in .
Now simplify the derivative by factoring the
numerator and canceling.
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What if the exponent on e is a function of x and
not just x?
Rule 2 If f(x) is a differentiable function
then
In words the derivative of e to the f(x) is an
exact copy of e to the f(x) times the derivative
of f(x).
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Example 3 Find the derivative of f(x)
Solution We will have to use Rule 2. The
exponent, 3x is a function of x whose derivative
is 3.
Times the derivative of the exponent
An exact copy of the exponential function
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Example 4 Find the derivative of
Solution
Again, we used Rule 2. So the derivative is the
exponential function times the derivative of the
exponent.
Or rewritten
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Example 5 Differentiate the function
Solution
Using the quotient rule
Keep in mind that the derivative of e-t is
e-t(-1) or -e-t
Recall that e0 1.
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Find the derivative of .
Click on the button for the correct answer.
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No, the other answer was correct. Remember when
you are doing the derivative of e raised to the
power f(x) the solution is e raised to the same
power times the derivative of the exponent.
What is the derivative of ?
Try again. Return
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Good work!! Here is the derivative in detail.
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Example 6 A quantity growing according to the
law where Q0 and k are
positive constants and t belongs to the
interval experiences exponential
growth. Show that the rate of growth Q(t) is
directly proportional to the amount of the
quantity present.
Solution
Remember To say Q(t) is directly proportional
to Q(t) means that for some constant k, Q(t)
kQ(t) which was easy to show.
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Example 7 Find the inflection points of
Solution We must use the 2nd derivative to find
inflection points.
First derivative
Product rule for second derivative
Simplify
Set equal to 0.
Exponentials never equal 0.
Set the other factor 0.
Solve by square root of both sides.
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To show that they are inflection points we put
them on a number line and do a test with the 2nd
derivative
-
Intervals Test Points Value
-1 0 1
f(-1) 4e-1-2e-12e-1 f(0)0-2-2
- f(1) 4e-1-2e-12e-1
Since there is a sign change across the potential
inflection points,
and are inflection points.
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In this lesson you learned two new rules of
differentiation and used rules you have
previously learned to find derivatives of
exponential functions. The two rules you
learned are
Rule 1 Derivative of the Exponential Function
Rule 2 If f(x) is a differentiable function
then