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
A restaurant manager can spend at most $600 a day for operating costs and payroll. It costs $100 each day to operate the bank an
navik [9.2K]

Option A

The restaurant Manager can afford at most 10 employees for the day

<em><u>Solution:</u></em>

Given that restaurant manager can spend at most $600 a day for operating costs and payroll

It costs $100 each day to operate the bank and $50 dollars a day for each employee

The given inequality is:

50x + 100\leq 600

Where , "x" is the number of employees per day

Let us solve the inequality for "x"

50x + 100\leq 600

Add -100 on both sides of inequality

50x + 100 - 100\leq 600 - 100\\\\50x\leq 500

Divide by 50 on both sides of inequality

\frac{50x}{50}\leq \frac{500}{50}\\\\x\leq 10

Hence the restaurant Manager can afford at most 10 employees for the day

Thus option A is correct

6 0
3 years ago
Angles H and F are corresponding angles<br> True or False?
lutik1710 [3]
True! they are corresponding angles.
5 0
1 year ago
Read 2 more answers
Last year, Dale mowed the lawn in 45 min. This year, it took Dale 38 min to mow the lawn. What is the percent of decrease in mow
klasskru [66]
45-38= 7min decrease
(7/45)* 100=15.5%
3 0
3 years ago
Find the value of X<br> PLEASE HELP ASAP!!!!!
MAVERICK [17]

Answer:

-10

Step-by-step explanation:

the upper right is the same (105°), therefore the left 75°

now you have 180 = 75 + x + 115

therefore: -10

6 0
3 years ago
Find the missing term. ...-45,__,-39...
Anika [276]
The answer is 32 because -45 + 3 = -32 and -39 - 3 = -32
8 0
3 years ago
Other questions:
  • A number k less than 10
    9·1 answer
  • LOTS OF POINTS! WILL GIVE BRAINIEST. <br> (no spam plz)<br> How would I answer these?
    15·2 answers
  • Write each fraction or mixed<br> number as a decimal and percent<br>2/5
    12·2 answers
  • A rectangular prism has a length of 16 feet, a width of 9 feet, and a height of 8 feet. Find the volume of the prism.
    15·1 answer
  • Determine if the two lines are parallel, perpendicular, or neither.<br><br> x=5 x=−5
    8·1 answer
  • Please help and show the work to this page !!
    10·2 answers
  • PLEASE HELP! Identify the rule for the function table.
    12·1 answer
  • What is the slope of (12,22) (-20,19)
    15·1 answer
  • Help !!!!!!!!!!!!!!!
    15·2 answers
  • Tyler has to savings accounts that his grandparents opened for him . One of the accounts Pay 8% annual interest Where as the oth
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!