<< Chapter < Page Chapter >> Page >
The functions f(x) = sin x and f’(x) = cos x are graphed. It is apparent that when f(x) has a maximum or a minimum that f’(x) = 0.
Where f ( x ) has a maximum or a minimum, f ( x ) = 0 that is, f ( x ) = 0 where f ( x ) has a horizontal tangent. These points are noted with dots on the graphs.

Differentiating a function containing sin x

Find the derivative of f ( x ) = 5 x 3 sin x .

Using the product rule, we have

f ( x ) = d d x ( 5 x 3 ) · sin x + d d x ( sin x ) · 5 x 3 = 15 x 2 · sin x + cos x · 5 x 3 .

After simplifying, we obtain

f ( x ) = 15 x 2 sin x + 5 x 3 cos x .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the derivative of f ( x ) = sin x cos x .

f ( x ) = cos 2 x sin 2 x

Got questions? Get instant answers now!

Finding the derivative of a function containing cos x

Find the derivative of g ( x ) = cos x 4 x 2 .

By applying the quotient rule, we have

g ( x ) = ( sin x ) 4 x 2 8 x ( cos x ) ( 4 x 2 ) 2 .

Simplifying, we obtain

g ( x ) = −4 x 2 sin x 8 x cos x 16 x 4 = x sin x 2 cos x 4 x 3 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the derivative of f ( x ) = x cos x .

cos x + x sin x cos 2 x

Got questions? Get instant answers now!

An application to velocity

A particle moves along a coordinate axis in such a way that its position at time t is given by s ( t ) = 2 sin t t for 0 t 2 π . At what times is the particle at rest?

To determine when the particle is at rest, set s ( t ) = v ( t ) = 0 . Begin by finding s ( t ) . We obtain

s ( t ) = 2 cos t 1 ,

so we must solve

2 cos t 1 = 0 for 0 t 2 π .

The solutions to this equation are t = π 3 and t = 5 π 3 . Thus the particle is at rest at times t = π 3 and t = 5 π 3 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

A particle moves along a coordinate axis. Its position at time t is given by s ( t ) = 3 t + 2 cos t for 0 t 2 π . At what times is the particle at rest?

t = π 3 , t = 2 π 3

Got questions? Get instant answers now!

Derivatives of other trigonometric functions

Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives.

The derivative of the tangent function

Find the derivative of f ( x ) = tan x .

Start by expressing tan x as the quotient of sin x and cos x :

f ( x ) = tan x = sin x cos x .

Now apply the quotient rule to obtain

f ( x ) = cos x cos x ( sin x ) sin x ( cos x ) 2 .

Simplifying, we obtain

f ( x ) = cos 2 x + sin 2 x cos 2 x .

Recognizing that cos 2 x + sin 2 x = 1 , by the Pythagorean theorem, we now have

f ( x ) = 1 cos 2 x .

Finally, use the identity sec x = 1 cos x to obtain

f ( x ) = sec 2 x .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the derivative of f ( x ) = cot x .

f ( x ) = csc 2 x

Got questions? Get instant answers now!

The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem.

Derivatives of tan x , cot x , sec x , And csc x

The derivatives of the remaining trigonometric functions are as follows:

d d x ( tan x ) = sec 2 x
d d x ( cot x ) = csc 2 x
d d x ( sec x ) = sec x tan x
d d x ( csc x ) = csc x cot x.

Finding the equation of a tangent line

Find the equation of a line tangent to the graph of f ( x ) = cot x at x = π 4 .

To find the equation of the tangent line, we need a point and a slope at that point. To find the point, compute

f ( π 4 ) = cot π 4 = 1 .

Thus the tangent line passes through the point ( π 4 , 1 ) . Next, find the slope by finding the derivative of f ( x ) = cot x and evaluating it at π 4 :

f ( x ) = csc 2 x and f ( π 4 ) = csc 2 ( π 4 ) = −2 .

Using the point-slope equation of the line, we obtain

y 1 = −2 ( x π 4 )

or equivalently,

y = −2 x + 1 + π 2 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Finding the derivative of trigonometric functions

Find the derivative of f ( x ) = csc x + x tan x .

To find this derivative, we must use both the sum rule and the product rule. Using the sum rule, we find

f ( x ) = d d x ( csc x ) + d d x ( x tan x ) .

In the first term, d d x ( csc x ) = csc x cot x , and by applying the product rule to the second term we obtain

d d x ( x tan x ) = ( 1 ) ( tan x ) + ( sec 2 x ) ( x ) .

Therefore, we have

f ( x ) = csc x cot x + tan x + x sec 2 x .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask