Ask a Question
Home
Take a Tour
Study Tips
Help Pages
Contact Us
Gallery
Recently Added
Videos
Search
Blogs
Science Blog
Fantastic Facts Blog
Academic Blog
Dictionary
Add New Word
Random Term
Recent Changes
Resource Library
Share Your Resources
Recently Added
Search
Forum
Ask a Question
Log In
Log In
Forgot your password?
Sign Up
for all your
science
needs
Forum
Search
Advanced Search
Image Gallery
Resource Library
Topics
Unresolved Questions
Discussions
Trending
Recent Posts
Chat
Subjects
Join over
250,000
students, educators, and subject enthusiasts in our free online study community
More
Q & A Board
Entire Website
Gallery
Resource Library
Dictionary
Blogs
Search through 1.7M homework questions
Unresolved Questions
•
Discussions
•
Recent
Top Posters
Since Sunday
27
25
23
21
18
18
18
18
18
18
17
17
Post a New Question
Current Questions
Home
Q & A Board
Science-Related Homework Help
Mathematics
Calculus
Print
New Topic
Finding the derivative of inverse tangent
bio_man
bio_man
wrote...
Administrator
Educator
Posts:
23128
Rep:
2887
9
3 months ago
3 months ago
Finding the derivative of inverse tangent
From the image, your function is: \(y=\frac{1}{4}\tan ^{-1}\left(\frac{4\sin x}{3+5\cos x}\right)\)
Start by multiplying both sides by \(4\):
\(4y=\tan ^{-1}\left(\frac{4\sin x}{3+5\cos x}\right)\)
That tangent of both sides:
\(\tan \left(4y\right)=\tan \left(\tan ^{-1}\left(\frac{4\sin x}{3+5\cos x}\right)\right)\)
This becomes:
\(\tan \left(4y\right)=\frac{4\sin x}{3+5\cos x}\)
Now differentiate implicitly with respect to \(x\). And recall that:
\(\frac{d}{dx}\tan u=\sec ^2u\cdot \frac{du}{dx}\)
Applying this to the left side ...
\(\sec ^2\left(4y\right)\cdot 4\ \frac{dy}{dx}=\frac{4\sin x\left(-5\sin x\right)-4\cos x\left(3+5\cos x\right)}{\left(3+5\cos x\right)^2}\)
Solve for \(\frac{dy}{dx}\):
\(\frac{dy}{dx}=\frac{4\sin x\left(-5\sin x\right)-4\cos x\left(3+5\cos x\right)}{4\ \sec ^2\left(4y\right)\left(3+5\cos x\right)^2}\)
Simplify the numerator:
\(\frac{dy}{dx}=\frac{-20\sin ^2x-12\cos x-20\cos ^2x}{4\ \sec ^2\left(4y\right)\left(3+5\cos x\right)^2}\)
Simplify more:
\(\frac{dy}{dx}=\frac{-20\left(\sin ^2x+\cos ^2\right)-12\cos x}{4\ \sec ^2\left(4y\right)\left(3+5\cos x\right)^2}\)
Recall that: \(\sin ^2x+\cos ^2x=1,\ \therefore \)
\(\frac{dy}{dx}=\frac{-5-3\cos x}{\sec ^2\left(4y\right)\left(3+5\cos x\right)^2}\)
Simplify more:
\(\frac{dy}{dx}=\frac{-1}{\sec ^2\left(4y\right)\left(3+5\cos x\right)}\)
There you go!
Read 138 times
Report
The best way to say thank you is with a positive review:
https://trustpilot.com/review/biology-forums.com
Your support goes a long way!
▶
Make a note request here
New Topic
Hold tight!
A
Biology Forums Expert
has been contacted to start answering this thread.
Share This Topic
Similar topics that might interest you...
Solved
Finding the slope of the tangent of a line?
Mathematics
6 years ago
irina
David Duong
1 Reply
373 Views
Solved
How would you go about finding the derivative of (e^(16x-5))*(5^(8x))?
Mathematics
6 years ago
ilohim
toni
1 Reply
195 Views
Solved
How do you find the derivative of the unit tangent vector?
Mathematics
5 years ago
tony123
rivertube
2 Replies
249 Views
Solved
Finding the derivative of a function two ways?
Mathematics
5 years ago
riya3rg
rivs9
1 Reply
244 Views
Any help with finding an inverse function?
Mathematics
5 years ago
toonoo
nurse2be
2 Replies
271 Views
Solved
Finding Tangent Lines: Finding equations at the points (6,0) & (0,0)
Mathematics
4 years ago
krutig
padre
6 Replies
362 Views
Finding the derivative of functions with tangent
Calculus
10 months ago
chementhusiast
bio_man
1 Reply
39 Views
Find the derivative of the inverse trig fn below
Calculus
2 weeks ago
Inverse trig fn
bio_man
1 Reply
33 Views
4 More
Quick Reply
[center][color=gray]Please [b]login or register[/b] to leave a reply[/color][/center]
Explore
Post your homework questions and get free online help from our incredible volunteers.
Why Sign Up?
Learn More
Improve Grades
Help Others
Save Time
Accessible 24/7
124 People Browsing
Start New Topic
Take the Tour
Study Tips
Browse by Textbook
New
Related Images
11018
68
55
Your Opinion
In one word, how would you rate our website?
Awesome
Good
Mediocre
Other
Votes: 64
Previous poll results:
Does a gender wage gap exist in America?
Latest Blogs
Animal longevity correlated to these genes
A case for why teenagers make risky decisions
A single molecule is reversing failing memories
Eyeless fish
Key energy-producing proteins missing in this plant
Ready to ask a question on
Biology Forums
?
Try it out
▸
Many children do not master phonology until they are eight or nine years old.
–
Education Studies
The most severe hearing impairments are associated with which part of the ear?
–
Education Studies
Most speech sounds have frequencies (pitch) that
–
Education Studies
Decoding, or understanding messages, is referred to as
–
Education Studies
More girls than boys stutter.
–
Education Studies
Jeremy is a four-year-old boy who makes infantile noises and uses gestures to request objects. He displays
–
Education Studies
Many children quickly outgrow their dysfluencies.
–
Education Studies
Methods of progress monitoring that involve a cycle of teaching, followed by testing, and then ...
–
Education Studies
Which of the following statements about written language development is TRUE?
–
Education Studies
A language disorder that involves problems with functional and socially appropriate communication is
–
Education Studies
Loading...