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Derivatives Using Definition Worksheet

Derivatives Using Definition Worksheet
  • Page 1
 1.  
Find the derivative of the function (y) = 9y + 14 directly from the definition of derivative.
a.
14
b.
9
c.
1 9
d.
- 9


Solution:

(y) = 9y + 14
[Write the function.]

′ (y) = limh0 f (y+h)-f (y)h
[Use the definition of derivative.]

= limh0 (9(y+h)+14)-(9y+14)h
[Use the definition of (y).]

= limh0 (9hh) = 9
[Evaluate the limit.]

′ (y) = 9


Correct answer : (2)
 2.  
Use the definition of the derivative to find the derivative of g(s) = 4s2 + 6s + 46.
a.
8s
b.
8
c.
4s + 6
d.
8s + 6
e.
4s


Solution:

g(s) = 4s2 + 6s + 46
[Write the function.]

g′ (s) = limh0 g(s+h)-g(s)h
[Use the definition of derivative.]

= limh0 (4(s+h)2+6(s+h)+46)-(4s2+6s+46)h
[Use the definition of g(s).]

= limh0 (4h2+8sh+6hh) = 8s + 6
[Evaluate the limit.]

g′ (s) = 8s + 6


Correct answer : (4)
 3.  
From the definition of derivative, find the derivative of (v) = 6v5/9 + 26
a.
5 9v - 4/9
b.
v - 4/9
c.
- 10 3v - 4/9
d.
54 5v 4/9
e.
10 3v - 4/9


Solution:

(v) = 6v5/9 + 26
[Write the function.]

′ (v) = limh0f (v+h)-f (v)h
[Use the definition of of ′ (v).]

= limh0(6(v+h)5/9+26)-(6v5/9+26)h
[Use the definition of (v).]

= 6(limh0(v+h)5/9 -v5/9h)

= 10 / 3v - 4/9
[Evaluate the limit.]

′ (v) = 10 / 3v - 4/9


Correct answer : (5)
 4.  
Use the definition of the derivative to find g′ (x) if g(x) = 11x9 + 15.
a.
9x8
b.
99x8
c.
99x9
d.
11x8
e.
99x8 + 15


Solution:

g(x) = 11x9 + 15
[Write the function.]

g′ (x) = limh0 g(x+h)-g(x)h
[Use the definition of the derivative.]

= limh0 (11(x+h)9+15)-(11x9+15)h
[Use the definition of g(x).]

= 11(limh0 (x+h)9-x9h)

= 11(9x8) = 99x8
[Evaluate the limit.]

g′ (x) = 99x8


Correct answer : (2)
 5.  
Find the derivative of (t) = 186t+27 using the definition of derivative.
a.
- 9 (6t + 27)- 3/2
b.
- 54 (6t + 27)- 3/2
c.
- 1 9(6t + 27)- 3/2
d.
(6t + 27)- 3/2
e.
- (6t + 27)- 3/2


Solution:

(t) = 186t+27
[Write the function.]

′ (t) = limh0 f (t+h)-f (t)h
[Use the definition of ′ (t).]

= limh0 (186(t+h)+27-186t+27h)
[Use the definition of (t).]

= 18 limh0 (6t+27-6(t+h)+27h (6(t+h)+27) (6t+27))

= - 54 (6t + 27)- 3/2
[Evaluate the limit.]

′ (t) = - 54 (6t + 27)- 3/2


Correct answer : (2)
 6.  
Find the derivative of g(v) = 11v + 13.
a.
11v2
b.
12v
c.
11v
d.
- 112v
e.
112v


Solution:

g(v) = 11v + 13
[Write the function.]

g ′ (v) = limh0 g(v+h)-g(v)h
[Use the definition of g′ (v).]

= limh0 (11v+h+13)-(11v+13)h
[Use the definition of g(v).]

= 11(limh0 v+h-vh)

= 11 limh0(1v+h+v) = 112v
[Rationalize the numerator and evaluate the limit.]

g ′ (v) = 112v


Correct answer : (5)
 7.  
Find the derivative of g(v) = 7tan 6v from the definition of derivative.
a.
42 sec 6v
b.
sec2 6v
c.
42 cosec2 6v
d.
42 sec2 6v
e.
1 42cos2 6v


Solution:

g(v) = 7tan 6v
[Write the function.]

g′(v) = limh0 g(v+h)-g(v)h
[Use the definition of g′ (v).]

= limh0 (7tan 6(v+h)-7tan 6vh)
[Use the definition of g(v).]

= 7 limh0 (sin 6hhcos 6(v+h)cos 6v)
[Use tan A - tan B = sin (A-B)cos A cos B]

= 7 (limh0 (sin 6hh)) (limh0 1cos 6(v+h)cos 6v)
[Use the Product Rule of limits.]

= (7) (6) 1(cos 6v)2 = 42 sec2 6v
[Use limx0 sin kxx = k and evaluate the limits.]

g′ (v) = 42 sec2 6v


Correct answer : (4)
 8.  
Find ′ (p), if (p) = 8cot 7p + 12 from the definition of derivative.
a.
1 56sin2 7p
b.
- 56 cosec2 7p
c.
56 cosec2 7p
d.
- 7 cosec2 7p
e.
- 8 cosec2 7p


Solution:

(p) = 8cot 7p + 12
[Write the function.]

′(p) = limh0 f (p+h)-f (p)h
[Use the definition of ′ (p).]

′ (p) = limh0 (8cot 7(p+h)+12)-(8cot 7p+12)h
[Use the definition of (p).]

= 8 limh0 cot 7(p+h)-cot 7ph

= 8 limh0 (sin (- 7h)h sin 7(p+h)sin 7p)
[Use cot A - cot B = sin (B-A)sin A sin B]

= (- 8) (limh0 sin 7hh) (limh0 1sin 7(p+h) sin 7p)
[Use the Product Rule of limits.]

= (- 8) (7) (1(sin 7p)2) = - 56 cosec2 7p
[Use limx0 (sin kxx) = k and evaluate the limits.]

′ (p) = - 56 cosec2 7p


Correct answer : (2)
 9.  
Find the derivative of (x) = 169x+ 12 using the definition of derivative.
a.
144(9x+12)2
b.
- 144(9x+12)2
c.
- 9(9x+12)2
d.
16 9(9x+12)2
e.
9(9x+12)2


Solution:

(x) = 169x+12
[Write the function.]

′ (x) = limh0f (x+h)-f (x)h
[Use the definition of '(x).]

= limh0(169(x+h)+12-169x+12h)
[Use the definition of (x).]

= 16limh0(9x+12)-(9(x+h)+12)h(9(x+h)+12) (9x+12)

= 16 limh0(- 9(9(x+h)+12) (9x+12))

= - 144(9x+12)2

′ (x) = - 144(9 x+12)2


Correct answer : (2)
 10.  
Calculate the derivative of (l) = 5l+813l+17 from the definition of derivative.
a.
19(13l+17)2
b.
-19(13l-17)2
c.
(13l+17)2189
d.
189(13l+17)2
e.
-19(13l+17)2


Solution:

(l) = 5l+813l+17
[Write the function.]

′ (l) = limh0f (l+h) - f (l)h
[Use the definition of ′ (l).]

= limh0(5(l+h)+813(l+h)+17-5l+813l+17h)
[Use the definition of (l).]

= -19(13l+17)2
[Evaluate the limit.]

′ (l) = -19(13l+17)2


Correct answer : (5)

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