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Derivatives of Logarithmic Functions Worksheet

Derivatives of Logarithmic Functions Worksheet
  • Page 1
 1.  
If g(l) = ln (sec 52l + tan 52l), then find g′ (l).
a.
52sec 52l+tan 52l
b.
52sec 52l
c.
1sec 52l+tan 52l
d.
sec 52l
e.
52sec 52l(sec 52ltan 52l)+52tan 52l(sec252l)


Solution:

g(l) = ln (sec 52l + tan 52l)
[Write the function.]

g′ (l) = ddl( ln (sec 52l + tan 52l))
[Find g′ (l)]

= 1sec 52l+tan 52lddl(sec 52l+tan 52l)
[Use the Chain Rule.]

= 52sec 52ltan 52l+52sec252lsec 52l+tan 52l

= 52sec 52l(tan 52l+sec 52l)sec 52l+tan 52l = 52sec 52l
[Factor out 52sec 52l and simplify.]

g′ (l) = 52sec 52l


Correct answer : (2)
 2.  
If g(q) = q10ln 8q, then find g′ (q).
a.
q9(18+10ln 8q)
b.
10q9(ln 8q)
c.
q9(18-10ln 8q)
d.
q9(1-10ln 8q)
e.
q9(1+10ln 8q)


Solution:

g(q) = q10ln 8q
[Write the function.]

g ′(q) = ddq(q10ln 8q)
[Find g ′(q).]

= q10ddq(ln 8q)+(ln 8q)ddq(q10)
[Use the Product Rule.]

= q10(1q)+(ln 8q)(10q9)
[Use the Chain Rule.]

= q9(1+10ln 8q)

g ′(q) = q9(1+10ln 8q)


Correct answer : (5)
 3.  
If (x) = x(ln 8x)11, then find ′ (x).
a.
(ln 8x)10(118 + (ln 8x))
b.
(ln 8x)10(118 - (ln 8x))
c.
(ln 8x)10(11 + (ln 8x))
d.
(10(ln 8x))(11 + (ln 8x))
e.
(ln 8x)10(11 - (ln 8x))


Solution:

(x) = x(ln 8x)11
[Write the function.]

′ (x) = ddx(x(ln 8x)11)
[Find ′ (x)]

= (x)ddx((ln 8x)11)+(ln 8x)11ddx(x)
[Use the Product Rule.]

= 11x(ln 8x)10ddx(ln 8x)+(ln 8x)11(1)
[Use the Chain Rule.]

= 11x(ln 8x)10(1x)+(ln 8x)11

= (ln 8x)10(11 + (ln 8x))

′ (x) = (ln 8x)10(11 + (ln 8x))


Correct answer : (3)
 4.  
If (k) = 7(ln 9k11+31), then find ′ (k).
a.
(693 2) (k109k11+31)
b.
(693 2) (k109k11+31)
c.
(k109k11+31)
d.
7 ln(99k1029k11+31)
e.
(7 9k11+31)


Solution:

(k) = 7(ln 9k11+31)
[Write the function.]

′ (k) = ddk(7ln9k11+31)
[Find ′ (k).]

= 7(19k11+31)ddk(9k11+31)
[Use the Chain Rule.]

= 7(19k11+31)(129k11+31)ddk(9k11+31)
[Use the Chain Rule again.]

= 7(9)(11)k102(9k11+31) = (693 / 2) (k109k11+31)

′ (k) = (693 / 2) (k109k11+31)


Correct answer : (2)
 5.  
If g (x) = ln (x + x2+8), then find g′ (x).
a.
ln(1 + xx2+8)
b.
1x + xx2+8
c.
1x2+8
d.
1x - xx2+8
e.
1x-x2+8


Solution:

g(x) = ln (x + x2+8)
[Write the function.]

g′(x) = ddx(ln(x+x2+8))
[Find g′(x).]

= 1x+x2+8ddx(x+x2+8)
[Use the Chain Rule.]

= 1x+x2+8(1 + (2x)2x2+8)
[Use the Chain Rule again.]

= 1x2+8

g′(x) = 1x2+8


Correct answer : (3)
 6.  
If g(t) = (ln (2t+6))11, then find g′ (t).
a.
22(ln (2t+6))10(2t+6)
b.
222 t+6
c.
112 t+6
d.
11(ln (2t+6))10(2t+6)
e.
22(ln (2t+6))10


Solution:

g(t) = (ln (2t+6))11
[Write the function.]

g′ (t) = ddt((ln (2t+6))11)
[Find g′ (t).]

= 11(ln (2t+6))10ddt(ln (2t+6))
[Use the General Power Rule and the Chain Rule.]

= 11(ln (2t+6))10(2(2t+6))
[Use the Chain Rule again.]

g′ (t) = 22(ln (2t+6))10(2t+6)


Correct answer : (1)
 7.  
If (w) = ln[cos (2w + 7)], then find ′ (w).
a.
- tan (2w + 7)
b.
- 2tan (2w + 7)
c.
tan (2w + 7)
d.
2tan (2w + 7)
e.
- 1cos(2w+7)


Solution:

(w) = ln[cos (2w + 7)]
[Write the function.]

′ (w) = ddw(ln[cos (2w+7)])
[Find ′ (w).]

= 1cos (2w+7)ddw(cos (2w+7))
[Use the Chain Rule.]

= - 2(sin (2w+7)cos (2w+7))
[Use the Chain Rule again.]

= - 2tan (2w + 7)

′ (w) = - 2tan (2w + 7)


Correct answer : (2)
 8.  
If g(u) = ln (8u+ 10)9u+12, then find (1440)g′ (0).
a.
96-90(ln 10)144
b.
90(ln 10) - 96
c.
96 - 90(ln 10)
d.
96 + 90(ln 10)
e.
90(ln 10)- 96144


Solution:

g(u) = ln (8u+10)9u+12
[Write the function.]

g′ (u) = ddu(ln (8u+10)9u+12)
[Find g′ (u).]

= (9u+12)ddu(ln (8u+10))-(ln (8u+10))ddu(9u+12)(9u+12)2
[Use the Quotient Rule.]

= (9u+12)(88u+10)-(ln (8u+10))(9)(9u+12)2

= 8(9u+12)-(ln (8u+10))(9)(8u+10)(8u+10)(9u+12)2

g′ (u) = 8(9u+12)-9(8u+10)(ln (8u+10))(8u+10)(9u+12)2

g′ (0) = 96-90ln 101440
[Find g′ (0).]

(1440)g′ (0) = 96 - 90(ln 10)


Correct answer : (3)
 9.  
If (l) = (ln (9l + 11))(ln (10l + 12)), then find ′ (0).
a.
(112) + (111)
b.
1132
c.
5 6(ln 11) + 9 11(ln 12)
d.
((ln 11)12) + ((ln 12)11)
e.
10(ln 9) + 9(ln 10)


Solution:

(l) = (ln (9l + 11))(ln (10l + 12))
[Write the function.]

′ (l) = ddl((ln (9l + 11))(ln (10l + 12)))
[Find ′ (l).]

= ln (9l + 11)ddl(ln (10l + 12)) + ln (10l + 12)ddl(ln (9l + 11))
[Use the Product Rule.]

= (ln (9l + 11))(1010l+12) + (ln (10l + 12))(99l+11)
[Use the Chain Rule.]

= 10(ln (9l + 11))10l+12 + 9(ln (10l + 12))9l+11

′ (0) = 5 / 6(ln 11) + 9 / 11(ln 12)
[Find ′ (0).]


Correct answer : (3)
 10.  
If (k) = ln(ln (6k + 7)), then find ′ (k).
a.
6(6k+7)(ln (6k+7))
b.
1k(ln (6k))
c.
1(6k+7)(ln (6k+7))
d.
6(6k+7)
e.
6ln (6k+7)


Solution:

(k) = ln(ln (6k + 7))
[Write the function.]

′ (k) = ddk(ln(ln (6k+7)))
[Find ′ (k).]

= 1ln (6k+7)ddk(ln (6k+7))
[Use the Chain Rule.]

= 6(6k+7)(ln (6k+7))

′ (k) = 6(6k+7)(ln (6k+7))


Correct answer : (1)

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