Differentiate ln(tanx)?

Question

Differentiate ln(tanx)

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Answer 1

f(x) = ln(tanx)
f '(x)
= 1/tanx * d/dx (tanx)
= 1/tanx * sec^2 x
= 1/(sinx/cosx) * 1/(cos^2x)
= cosx/sinx * 1/(cos^2x)
= cosx / (sinx)*(cos^2 x)
= 1/sinx cosx (cancelling cosx from numerator and denominator)
= 2cosec 2x

Answer 2

Ln Tan X

Answer 3

(d/dx) ln(tanx)
= (1/tanx) . (secx)^2
= (cosx/sinx) . [1/(cosx)^2]
= 1/(sinxcosx)
= cosecx secx

Answer 4

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We've been told we have to implement a new way of teaching in our reading classes that are centers/stations driven. I'm currently giving it a shot, and I see a lot of great things coming from it. As I continue using it, I'll tweak it to fit my style of teaching, but for now, I'm trying to stick to the original intention. Anyway, here's how it works. The class is divided into smaller homogeneous groups. Since you've got below level students, I'd stick to 3 groups and no more than that. That will make it easier for you to make a schedule. Each group rotates around three different things- skills stations, sustained silent reading (SSR), and fluency reading w/the teacher. During the skills stations, you've got all kinds of different activities that can make reading more hands on. You have to teach each station to the whole group before you can expect them to be able to do it independently, but if you make them generic so they can be used for any book, you won't have that many new things to teach. I've got a Fact/Opinion board, a Connections board (Text to Text connections, Text to Self connections, and Text to World connections), a Predict-o-gram board (you give them characters, setting, problem, solution, etc. and let the students place them under the headings in which they think they belong--or--you let them do a picture preview of the story and let them fill in what goes under each heading themselves), and...ah shoot, I've got a couple other boards that I haven't used yet, so I don't remember what they are. I also include a vocabulary station, but I change up what they do. Last week they played Memory with the words and definitions, this week they're writing an acceptance speech as president using the vocab words. If you teach writing and language arts along with reading, you can always have those stations as well. I've been making up my own stations too that are specific to the story we're working on. This week I've got Main Idea graphic organizers. The trick to all the stations is that you provide lots of sticky notes for them to "manipulate" what they're writing. For whatever reason, they're a lot more willing to write on stickies than they are in a notebook. However, they each have a "Reading Stations" notebook (they decorate the cover) that they transfer their stickies into. It makes it easy for me to grade them and/or assess where they're at. The SSR is the easiest one since it's just kids reading what THEY want to read. During the last 5 minuts of that rotation, the kids respond to what they've read in their stations notebook with sentence starters ("I was able to picture in my mind..." "I wonder why..." "I made a connection when"). The part with the teacher is the one that I think I'll personally tweak some, but the idea is that the kids focus on their fluency. The round-robbin way of reading does nothing for reading fluency/comprehension. Instead, the kids whisper-read something they haven't read before (but that is on their level). You go around and tap them on the shoulder, at which point they begin to read out loud. You take notes on how they're reading (self-correcting, sounding words out, finger following, too fast, too slow, etc.). Once you tap them again, they go back to whisper-reading. You're supposed to take the notes and use them to help you teach mini-lessons in those small groups later on (I think, lol). For example, I noticed that a couple of my students skipped an entire page when they were reading b/c it was a time line. I was able to do a quick mini-lesson on how to read time lines. I also noticed that almost none of them read headings or captions, so again, a quick mini-lesson. Anyway, so far I like it b/c I'm able to do small group instruction and differentiate with the mini-lessons. I also like that there's some silent reading built in so that they're reading what they want. We're still practicing how to whisper and travel quietly between the stations, and we're still learning how to manage our time. I use a timer to help us all keep track, and I'm going to start using a musical triangle (it was the only thing the music teacher would let me have, lol) to help them know when their transition should be over. I should let you know that I teach 4th graders, so I'm sure that if you tried something like this that you'd have to model it a LOT more (especially the whispering). I've got 2 big posters that remind them what the stations are for the week. We created an expectations poster (whisper if you must, rotate quietly, refer to the poster if you have questions, etc.). I've also go the schedule posted. I named my groups after the 4 Beatles members. It was easy for me to remember which group was which w/o the kids recognizing which group was the highest and lowest. My only beef with it so far is that I don't feel the need to take notes on them 4 days a week, and I don't feel like there's enough whole-group instruction. We only

Answer 5

let u = tan x
du/dx = sec ² x
y = ln u
dy/du = 1 / u
dy/dx = (1/u) (sec ² x)
dy/dx = (1 / tan x) (sec ² x)
dy/dx = (cos x / sin x) (1 / cos ² x)
dy/dx = 1 / (sin x cos x)
dy/dx = cosec x sec x