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Exponential Functions Worksheet

Exponential Functions Worksheet
  • Page 1
 1.  
Find the derivative of f(x) = sin 3xcos 4x + 3e2x + 6x3
a.
18x2+6e2x+3cos 3xsin 4x
b.
18x2+6e2x+3cos 3xcos 4x-4sin 3xsin 4x
c.
18x2+6e2x-12sin 4xcos 3x
d.
18x2-6e2x-4sin 4xsin 3x


Solution:

f(x) = 3e2x+6x3+sin 3xcos 4x

f′(x) = ddx[3e2x+6x3+sin 3xcos 4x]
[Differentiate with respect to x on both sides.]

= 18x2+6e2x+3cos 3xcos 4x-4sin 3xsin 4x


Correct answer : (2)
 2.  
Find the derivative of h(x) = ecot22x + 2x3.
a.
(- 4cot 2xcosec22x + 6x2)ecot22x + 2x3
b.
(4cot 2x 6x2 + 2x3)
c.
(- cot 2xcosec22x + 2x3)
d.
(cosec22x + 6x2)ecot22x


Solution:

h(x) = ecot22x + 2x3

h′(x) = ddx[ecot2 2x + 2x3]
[Differentiate with respect to x on both sides.]

= ecot22x + 2x3ddx(cot22x + 2x3)
[Use dydx(ekx) = kekx.]

= (- 4cot 2xcosec22x + 6x2)ecot22x + 2x3


Correct answer : (1)
 3.  
Find the derivative of g(x) = e37+ cos 6x.
a.
 sin 6x37+cos 6x
b.
- 3  sin 6x37+cos 6xe37+cos 6x
c.
- sin 6x37+cos 6xe37+cos 6x
d.
- sin 6x37+cos 6x


Solution:

g(x) = e37+cos 6x

g′(x) = ddx[e37+cos 6x]
[Differentiate with respect to x on both sides.]

= e37+cos 6xddx(37+cos 6x)

= - 3  sin 6x37+cos 6xe37+cos 6x
[Use dydx(kx) =k2kx.]


Correct answer : (2)
 4.  
Find the derivative of g(x) = (e4x + e5x + e6x)19x.
a.
19(e4x(1+4x)+e6x(1+6x))
b.
(e4x(1+4x)-e5x(1+5x))
c.
19(e4x(1+4x)+e5x(1+5x)+e6x(1+6x))
d.
(e4x(1-4x)+e5x(1-5x)(1+6x))


Solution:

g(x) = (e4x+e5x+e6x)19x

g′(x) = (e4x+e5x+e6x)ddx(19x)+19xddx(e4x+e5x+e6x)
[Use the product rule.]

= 19(e4x+e5x+e6x)+19x(4e4x+5e5x+6e6x)

= 19(e4x[1+4x]+e5x[1+5x]+e6x[1+6x])
[Take out 19 as common factor.]


Correct answer : (3)
 5.  
Find the derivative of h(x) = ecos 7x + esin 6x.
a.
7 2- cos 7xsin 7x
b.
7 2- sin 7xcos 7x
c.
- 3 esin 6x
d.
7 2- sin 7xcos 7x.ecos 7x + 3 cos 6xsin 6x.esin 6x


Solution:

h(x) = ecos 7x + esin 6x

h′(x) = ddx[ ecos 7x] + ddx[esin 6x]

= ecos 7x.ddx[cos 7x] + esin 6x.ddx[sin 6x]

= 7 / 2- sin 7xcos 7x.ecos 7x + 3 cos 6xsin 6x.esin 6x


Correct answer : (4)
 6.  
Find the derivative of f(x) = 29ee2x.
a.
58ee2x
b.
58e2xee2x
c.
29e2xee2x
d.
2e2xee2x


Solution:

f(x) = 29ee2x

f′(x) = ddx(29ee2x)
[Differentiate with respect to x on both sides.]

= 29e2x(2)ee2x
[dydx(ekx) = kekx.]

= 58e2xee2x
[58 = 29 × 2.]


Correct answer : (2)
 7.  
Find the derivative of y = 40e- 5x cos 6x.
a.
- e- 5x [240sin 6x + cos 6x]
b.
- e- 5x [240sin 6x - 200cos 6x]
c.
- e- 5x [240sin 6x + 200cos 6x]
d.
- e 5x [240sin 6x + 200cos 6x]


Solution:

y = 40e - 5x cos 6x

dydx = 40ddx[ e - 5x cos 6x]
[Differentiate with respect to x on both sides.]

= 40e- 5x ddx(cos 6x) + 40(cos 6x) ddx( e- 5x)
[Use product rule.]

= - 240e- 5xsin 6x - 200e- 5x cos 6x
[Use cos kx = - ksin kx , dydx(ekx) = kekx.]

= - e- 5x [240sin 6x + 200cos 6x]


Correct answer : (3)
 8.  
Find dydx if y = 40e5x log sin 4x.
a.
40ex [4tan 4x + 5log(sin 4x)]
b.
40e5x [4cot 4x + 5log(sin 4x)]
c.
40ex [4cot 4x - 5log(cos 4x)]
d.
40e5x [4tan 4x - 5log(cos 4x)]


Solution:

y = 40e5xlog sin 4x.

dydx = ddx[40e5x log(sin 4x)]
[Differentiate both sides with respect to x.]

= 40[e5x ddx(log(sin 4x)) + log(sin 4x)ddx(e5x)]
[Use product rule.]

= 40[4e5x cot 4x + 5e5x log(sin 4x)]
[Use log(sin x) = cot x.]

= 40e5x [4cot 4x + 5log(sin 4x)]


Correct answer : (2)
 9.  
Find the derivative of f(x) = 46sin2(e4x).
a.
184sin 2e- 4x
b.
184e4xsin 2e4x
c.
184e4xsin e- 4x
d.
184e4xcos e4x


Solution:

f(x) = 46sin2(e4x)

f′(x) = ddx[ 46sin2(e4x)]
[Differentiate with respect to x on both sides.]

= 92sin(e4x)cos(e4x)ddx(e4x)
[Use chain rule.]

= 368e4xsin(e4x)cos(e4x)
[Use chain rule.]

= 184e4x [2sin(e4x)cos(e4x)]
[Use 368 = 2×184.]

= 184e4xsin 2e4x
[Use 2sin θcos θ = sin 2θ.]


Correct answer : (2)
 10.  
Find the derivative of h(x) = 8x2e3x.
a.
8xe3x(3x + 2)
b.
xe3x(x + 2)
c.
xe3x(x² - 2)
d.
e3x(x² + 16)


Solution:

h(x) = 8x2e3x

h′(x) = ddx[8x2e3x]
[Differentiate with respect to x on both sides.]

= ddx(8x2e3x)
[Use product rule.]

= [8x2ddx(e3x) + e3xddx(8x2)]
[Use chain rule.]

= [8x2(3e3x) + e3x16x]
[Use dydx(ekx) = kekx.]

= 8xe3x[3x + 2]
[Take out 8xe3x as common factor.]


Correct answer : (1)

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