1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
galina1969 [7]
3 years ago
6

Find the derivative of ln(secx+tanx)

Mathematics
1 answer:
Sliva [168]3 years ago
3 0
If you're using the app, try seeing this answer through your browser:  brainly.com/question/3000160

————————

Find the derivative of

\mathsf{y=\ell n(sec\,x+tan\,x)}\\\\\\ \mathsf{y=\ell n\!\left(\dfrac{1}{cos\,x}+\dfrac{sin\,x}{cos\,x} \right )}\\\\\\ \mathsf{y=\ell n\!\left(\dfrac{1+sin\,x}{cos\,x} \right )}


You can treat  y  as a composite function of  x:

\left\{\! \begin{array}{l} \mathsf{y=\ell n\,u}\\\\ \mathsf{u=\dfrac{1+sin\,x}{cos\,x}} \end{array} \right.


so use the chain rule to differentiate  y:

\mathsf{\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot \dfrac{du}{dx}}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{d}{du}(\ell n\,u)\cdot \dfrac{d}{dx}\!\left(\dfrac{1+sin\,x}{cos\,x}\right)}


The first derivative is  1/u, and the second one can be evaluated by applying the quotient rule:

\mathsf{\dfrac{dy}{dx}=\dfrac{1}{u}\cdot \dfrac{\frac{d}{dx}(1+sin\,x)\cdot cos\,x-(1+sin\,x)\cdot \frac{d}{dx}(cos\,x)}{(cos\,x)^2}}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{1}{u}\cdot \dfrac{(0+cos\,x)\cdot cos\,x-(1+sin\,x)\cdot (-\,sin\,x)}{(cos\,x)^2}}


Multiply out those terms in parentheses:

\mathsf{\dfrac{dy}{dx}=\dfrac{1}{u}\cdot \dfrac{cos\,x\cdot cos\,x+(sin\,x+sin\,x\cdot sin\,x)}{(cos\,x)^2}}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{1}{u}\cdot \dfrac{cos^2\,x+sin\,x+sin^2\,x}{(cos\,x)^2}}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{1}{u}\cdot \dfrac{(cos^2\,x+sin^2\,x)+sin\,x}{(cos\,x)^2}\qquad\quad (but~~cos^2\,x+sin^2\,x=1)}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{1}{u}\cdot \dfrac{1+sin\,x}{(cos\,x)^2}}


Substitute back for  \mathsf{u=\dfrac{1+sin\,x}{cos\,x}:}

\mathsf{\dfrac{dy}{dx}=\dfrac{1}{~\frac{1+sin\,x}{cos\,x}~}\cdot \dfrac{1+sin\,x}{(cos\,x)^2}}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{cos\,x}{1+sin\,x}\cdot \dfrac{1+sin\,x}{(cos\,x)^2}}


Simplifying that product, you get

\mathsf{\dfrac{dy}{dx}=\dfrac{1}{1+sin\,x}\cdot \dfrac{1+sin\,x}{cos\,x}}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{1}{cos\,x}}


∴     \boxed{\begin{array}{c}\mathsf{\dfrac{dy}{dx}=sec\,x} \end{array}}\quad\longleftarrow\quad\textsf{this is the answer.}


I hope this helps. =)


Tags:  <em>derivative composite function logarithmic logarithm log trigonometric trig secant tangent sec tan chain rule quotient rule differential integral calculus</em>

You might be interested in
Is 936 divisible by 9?
VARVARA [1.3K]

Answer:

Yea

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
A drawer contains 4 white socks, 5 black socks, and 3 brown socks. Find the probability that...
IRINA_888 [86]

Answer:

5/12

7/12

125/36 = 3,47%

50/11 = 4,54%

Step-by-step explanation:

Probability a black sock is selected when a person chooses 1 sock = 5/12

Probability a white or brown sock is selected when a person chooses 1 sock =

7/12

Probability  a person chooses 3 socks and selects a white first, a black second, and a brown last if the socks are replace = (4/12 * 5/12 * 3/12)*100 =125/36 = 3,47%

Pobability  a person chooses 3 socks and selects a white first, a black second, and a brown last if the socks are NOT replace = (4/12 * 5/11 * 3/ 10)*100 = 50/11 = 4,54%

3 0
3 years ago
How to write 100,203 in standard form
True [87]
This is standard form
8 0
3 years ago
Read 2 more answers
Liz and Alana each participate in the home that jump at the track meet Liz’s high jump was 1 meter Alana’s high jump was 132 cen
brilliants [131]
1 meter is 100 centimeters, therefore Alana jumped higher by 32 centimeters.
132-100=32
7 0
3 years ago
Every night Nathan has to read at least 25 minutes. Write an inequality that shows how long Nathan can read each night
DaniilM [7]

Answer:

25<=x

i need 20 characters to answer

3 0
3 years ago
Read 2 more answers
Other questions:
  • Formulate a system of equations for the situation below and solve. Michael Perez deposited a total of $2000 with two savings ins
    13·2 answers
  • What is a number increased by 15 is 38
    9·1 answer
  • How would you find the least possible number of row without having to check if 8 is a factor of every between 70 and 150
    11·2 answers
  • To find out approximately how many hours its residents are spending playing cards at a senior retirement home, the recreation di
    5·1 answer
  • William read 15 books in 6 months. What was his rate of reading in months per book? {pls i need help}
    15·1 answer
  • From a standard deck of 52 cards, what is the probability of picking a 5 of diamonds at random from the deck?
    11·2 answers
  • A rectangle has an area of 12 square feet and a length of 5 feet. What is its width?
    5·2 answers
  • Help me out please thanks
    10·2 answers
  • HELP ME NO RANDOM ANSWERS
    13·1 answer
  • Offering Brainliest: Help with math!
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!