Deriv Probs

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Derivatives - Problems and Solutions

Laurel Benn

January 15, 2010

This problem and solution set should be an invaluable resource to dealwith the problems we have encountered in derivatives so far. I urge you tospend more time reading your notes, going over the problems we worked inclass, and working additional problems. Remember there is no substitutefor hard work.

Example 1 If f (x) =√

5x2 + 1, use the definition of the derivative to find

f ′(x).

Solution: f ′(x) = limh→0

f (x + h) − f (x)

h= lim

h→0

5(x + h)2 + 1 −

√ 5x2 + 1

hNext multiply both numerator and denominator by

5(x + h)2 + 1 +√

5x2 + 1 to get

limh→0

5(x + h)2 + 1 − (5x2 + 1)

h(

5(x + h)2 + 1 +√

5x2 + 1)= lim

h→0

5(x2 + 2xh + h2) + 1 − 5x2 − 1

h(

5(x + h)2 + 1 +√

5x2 + 1)

= limh→0

10xh + h2

h(

5(x + h)2 + 1 +√

5x2 + 1) = limh→0

10x + h 5(x + h)2 + 1 +

√ 5x2 + 1

= 10x2√ 5x2+1

= 5x√ 5x2+1

For other similar problems the setup is the same but we may have to do

a different algebraic manipulation to get what we want.

In the next example, I am assuming that we remember from class that(ln x)′ = 1

x.

Example 2 Find the derivative of f (x) = log7 x.

Solution:

f (x) = log7 x = ln xln 7 , by the change of base formula.

1



⇒ f ′(x) = 1ln 7

(lnx)′, since ln 7 is a constant.

⇒ f ′(x) = 1ln 7

· 1x

= 1x ln 7

In fact for any base a, (loga x)′ = 1x ln a

.

In class I showed you another way to solve this type of problem. Please refer

to your class notes.

Example 3 Find the derivative of y = x3 ln 2xex sinx

Solution:Take natural log of both sides of the equation to get:

ln y = ln(x3 ln 2x

ex sinx)

⇒ ln y = ln(x3 ln 2x) − ln(ex sin x)

⇒ ln y = lnx3 + ln(ln 2x) − ln ex − lnsinx

⇒ 1ydydx

= 3x2

x3+ 1

x ln 2x− 1 − cosx

sin x

⇒ 1

y

dy

dx=

3

x+

1

x ln 2x− 1 − cotx

⇒ dy

dx= y(

3

x+

1

x ln 2x− 1 − cotx)

⇒ dy

dx=

x3 ln 2x

ex sin x(

3

x+

1

x ln 2x− 1 − cotx)

This problem could also have been done with the quotient rule.

Example 4 Find the derivative of y = e2x cos3x√ (x−4)

Solution:

Take natural log of both sides of the equation to obtain:

ln y = ln e2x + ln cos3x− ln(√ x− 4)

⇒ ln y = 2x + ln cos3x− 12 ln(x− 4).

Differentiating both sides of the equation we get:1y

dy

dx= 2 + (−3sin3x)

cos3x− 1

21

x−4⇒ dy

dx= y(2 − 3tan3x− 1

2(x−4)

⇒ dy

dx= e2x cos3x√

x−4(2 − 3tan3x− 1

2(x−4)).

2



Example 5 Find the derivative of y =√

3x2 + 4x− 1.

y =√

3x2 + 4x− 1 = (3x2 + 4x− 1)1

2

dy

dx=

1

2(3x2 + 4x− 1)−

1

2 (6x + 4) =3x + 2√

3x2 + 4x−

1

Example 6 Find the derivative of y = 5x4 + 4x− 12x2 + 1√

x− 3.

y = 5x4 + 4x− x−2

2+ x−

1

2 − 3

dy

dx= 20x3 + 4 + x−3 − 1

2x−

3

2

= 20x3 +1

x3− 1

2√ x3

+ 4

Example 7 Find the derivative of f (x) = ln x3 .

f ′(x) = (ln x3)′ = (3ln x)′ = 3x

.

Example 8 If f (x) = ln[ln(x4 + 1)]. Find f ′(x).

Using the chain rule we get,

f ′(x) =1

ln(x4 + 1)· (ln(x4 + 1))′

=1

ln(x4 + 1)· 1

x4 + 1· 4x3

= 4x3

(x4 + 1) ln(x4 + 1)

Example 9 For the function f (x) = e−x2

2 , compute the first, second and

third derivatives of f (x).

f ′(x) = e−x2

2 · −2x

2= −xe−

x2

2

f ′′(x) = −x(−xe−x2

2 ) + e−x2

2 · (−1)

⇒ f ′′(x) = e−x2

2 (x2 − 1) = (x− 1)(x + 1)e−x2

2

f ′′′(x) = 2xe−x2

2 + −xe−x2

2 (x2 − 1) = x(3 − x2)e−x2

2

3



Example 10 Compute the derivative of y = 4π2.

Since π is a constant ⇒ 4π2 is a constant.

⇒ dy

dx= 0

Example 11 Differentiate y = sin x + 10 tanx w.r.t x.d

dx(sinx + 10 tanx) = cos x + 10sec2 x

Example 12 Differentiate y = xcosx

.

By the quotient rule we get,

x

cosx

′=

cosx(1) − x(− sin x)

cos2 x=

cosx + x sin x

cos2 x

Example 13 Differentiate y = sin(x cos x).

dy

dx = cos(x cos x) · (cosx− x sinx)

Example 14 For f (x) = ex−e−xex+e−x

. Find f ′(x).

Using the quotient rule we get ,

f ′(x) =(ex + e−x)(ex + e−x) − (ex − e−x)(ex − e−x)

(ex + e−x)2

=(ex + e−x)2 − (ex − e−x)2

(ex + e−x)2

=(ex + e−x + ex − e−x)(ex + e−x − ex + e−x)

(ex + e−x)2

=(2ex)(2e−x)

(ex + e−x)2

=4

(ex + e−x)2

Example 15 Find dydx

for y =√

1 + x1234.

dy

dx=

1

2√

1 + x1234· (1234x1233)

=

617x1233

√ 1 + x1234

4



Example 16 For f (x) = x ln x− x. Find f ′(x).

f ′(x) = ln x + x

1

x

− 1

= ln x

5

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