Formulas for Derivatives of Exponential Functionsmayaj/m142_Chapter4_Sec4.4complet… · 4 Spring...
5
Section 4.4 Derivatives of Exponential and Logarithmic Functions Formulas for Derivatives of Exponential Functions If f (x)= e x , then f 0 (x)= e x If f (x)= a x , where a 6= 0 is any real number, then f 0 (x)= a x · ln a If f (x)= e g(x) , then f 0 (x)= e g(x) · g 0 (x) If f (x)= a g(x) , then f 0 (x)= a g(x) · g 0 (x) · ln a 1. Find the derivative of the following. (a) f (x)=7 10x (b) f (x)=4 (x 7 +x) (c) y = e (5x 5 -4x) . hee = ex . Cl ) - - , Eat - - - Extra Fess - - = IF mess ' - = and . mass ' . Inna = y mess f ' ( x ) = 7mm . mess ' . he 77 ftp.719.fiol.hr#y=4meesf4xI=4mM.meeslehr4 f4×y=4""".(z×e+Dehr4Jf mass = e y ' = ends . mess ' ylaek×%4×).(zs×4_4)÷
Transcript of Formulas for Derivatives of Exponential Functionsmayaj/m142_Chapter4_Sec4.4complet… · 4 Spring...
Section 4.4 Derivatives of Exponential and Logarithmic Functions
Formulas for Derivatives of Exponential Functions
If f(x) = ex, then f 0(x) = ex
If f(x) = ax, where a 6= 0 is any real number, then f 0(x) = ax · ln a
If f(x) = eg(x), then f 0(x) = eg(x) · g0(x)
If f(x) = ag(x), then f 0(x) = ag(x) · g0(x) · ln a
1. Find the derivative of the following.
(a) f(x) = 710x
(b) f(x) = 4(x7+x)
(c) y = e(5x5�4x)
. hee = ex . Cl )- -
,
Eat- - - Extra
Fess - -
=IF mess
'
-
=and .
mass'
. Inna
= ymess
f'( x ) =
7mm . mess'
.he 77
ftp.719.fiol.hr#y=4meesf4xI=4mM.meeslehr4
f4×y=4""".(z×e+Dehr4Jfmass
= e
y'
=ends . mess
'
ylaek×%4×).(zs×4_4)÷
(d) f(x) = x4 · 5(x7+2)
(e) f(x) =p5 + 4e4x
Formulas for Derivatives of Logarithmic Functions
If f(x) = ln x, then f 0(x) =1
x
If f(x) = loga x, then f 0(x) =1
x · ln a
If f(x) = ln(g(x)), then f 0(x) =g0(x)
g(x)
If f(x) = loga(g(x)), then f 0(x) =g0(x)
g(x) · ln a
2 Spring 2019, Maya Johnson
."t'
-sues
9¥hit ? ( ¥m5
flickr.sk?tT#s'.x.lu5.x4J= mess
42
Tees
f'ha =
'
z. mess
' " ? mess'
f' I x , a
'zl5t4e49-":( 4e4x .
4)✓ I
f4xl=8l5t4e4×J"?e4xJ
-
-
←extra
-
-
= ¥- ness
Incas )
÷mass . In a
2. Find the derivative of the following.
(a) f(x) = log15(7� x4)
(b) g(w) = 3 ln(5 + 4w + w4)
(c) f(x) = (ln(1 + ex))5
(d) f(x) =x5
ln(9x4 + 5)
3 Spring 2019, Maya Johnson
= log, ,
( onees )
f 'Cxs=s7fka=i=3 in L mess )
gunI
3 . #
gi↳=s= mess
'S
Ess
f' ↳ =
5 . mess
".
mess'
ftp.5/lnCltexDT#eyes ,
suit a:÷⇒Hxi¥
3. Given F (x) = ln
✓x2(x2 + 4)5
(x2 � 2)4
◆.
(a) Use properties of logarithms to rewrite F (x) in terms of the sum and/or di↵erence of simpler
logarithms. Simplify as far as possible.
(b) Find the derivative of F(x).
4. Find an equation of the tangent line to the curve at the point (1, 1).
y = ln⇣xex
8⌘
4 Spring 2019, Maya Johnson
then = In ( x 4×2+435 ) - In ( x'
- 2)4
=@x¥xztus⑤_④.
Hx)=2mxt5mCx2t4)-4mCx2_IYH-e2-t5.e-4.TW#zfyxzXz
-
⇒ y = heyt
¥78-
⇒ y - lux t ×8 7 y
- I = 9×-9
je'
-
- ly t 8×7 y= 9 x - 9 t I
Slope= ly t
845 = 9-rE;Yy=o
5. Determine the value(s) of x for which f 0(x) = 0 for the following function.
f(x) = (9x+ 51)e3x+8
6. If $1, 200 is invested in a savings account o↵ering interest at a rate of 4.5% per year, compounded
continuously, how fast is the balance growing after 7 years? (Round your answer to the nearest
cent.)
5 Spring 2019, Maya Johnson
-3 x +8
(9xt5D#9 t 63×+8 ) ( 3)
fan -
. 9e+7?e¥sDf' ( x ) =3 e
3*8 ( 3 t 9×+51 )± 3 e3xt8( 9 x t 54 ) .
f ' C x ) = 0 ←→ 33*(9×+54)=0
32×+8=10 ⇒ l96t5Y# If-55¥ ⇒ x
- -
Find A'
(7)
A Lt ) =P ert ⇒ A Lt ) =L zoo £045 t
A'
It ) = 1200 . 645 ) . e.045T
V.
⇒ Att ) = 54 e.045T
A' l 7) = 54 e.
0450 =23.99$/yeaafter7yee
