Derivatives of Exponential
Functions
Base e is the most convenient base to use,
because it has the following property ……
d x
x
e =e
dx
Yes, this function is its own derivative! Let s
look at how this is true. First, let s recall the
limit definition of the derivative.
f '( x ) = limΔx→0
f ( x + Δx ) − f ( x )
Δx
Any exponential function can be turned
into a base e exponential function
Recall:
ln a = b..... means... e = a
b
so...( substitute_ out _ b)
e
ln a
=a
So…
( )
a = e
x
ln a x
a =e
x
x ln a
Let s apply the limit definition to the base e
exponential function.
f '( x ) = limΔx→0
f ( x + Δx ) − f ( x )
Δx
Now, let s recall the
definition of e and
include it in the formula.
e x ( 1 + Δx − 1)
f '( x ) = limΔx →0
Δx
Exponent cancels out
e x + Δx − e x
f '( x) = limΔx→0
Δx
e x (e Δx − 1)
f '( x) = limΔx→0
Δx
e = lim !x"0 (1 + !x )
1
!x
e Δx
x
f '( x) = limΔx→0
=e
Δx
x
Differentiating Base e functions
d u
e = u 'eu →" chain _ rule"
dx
d 2 x −1
e
= 2e 2 x −1
dx
You do this one………..
d " !3x %
e '=
$
dx # &
d " !3x % ( 3 + !3x
e '=* 2-e
$
dx # & ) x ,
!3
x
3e
= 2
x
Finding the Relative Extrema of a function
with a base e exponential
x
f
(
x
)
=
xe
Find the relative extrema of ….
Remember, relative extrema possible
where f (x) = 0 or f (x) DNE
Factor out the common
d
d
!
$
!
$
f '(x) = # ( x )& e x + #
ex & x
term
" dx %
" dx
%
f '(x) = e x + xe x
f '( x) = e x (1 + x)
( )
( )
Derivate exists (no
discontinuity). Set the
derivative equal to zero
0 = e (1 + x )
x
only _ when _ x = −1
0 = e (1 + x )
x
Check the answer graphically
First _ Derivative _ Test
f ' ( −2) = negative
f ' (0) = positive
so, the _ extrema _ is _ a _ MINIMUM _ POINT
only _ when _ x = −1
c
so... pt ._ is_ −1,−e−1
h
Remember …..
The derivative at a specific x-value tells you the slope
of the curve at that point on the curve.
the _ slope _ of _ f ( x ) = xe _ at _ x = 0
x
equals_ f '(0)
f '(0) = e (1 + 0) = 1(1) = 1
0
Problems for you……..
From page 369, Do #1-29 odds