find-the-derivative-simplify-where-possible-hxln-cosh-x-2

Question

Find the derivative. Simplify where possible.
$$
h(x)=\ln (\cosh x)
$$

Amy Jiang · Accepted Answer

Okay, so we have to find a derivative of h. Of X. So that&#x27;s H comma Rex. We&#x27;ll use our general here. So starting with Ellen, that&#x27;s one over our inside, which is coach of X and in times, interpretive over inside. So that&#x27;s equal to singer fax to send you over coached, that is, it will do hyperbolic tensions of X.

Doruk Isik · Answer

this from Where&#x27;s fund dirigible coastline immerse Be off B plus acres on X, divided by eight people&#x27;s connects him. You know that they were focused on your stuff or negative order was scared. Apartments and sperm. So then we have my front as native born forward skirt off. Born minus B plus acre Sine X, divided by a plus B Cups. I next squared months fly by director to DDX off speed plus a co sign X You want a plus B coaster X for we can write. Why prime as negative one Over are screwed off a plus. Speak of sign of X squared minus B plus a co sign of X squared divided by a plus B co signer. X squared. There&#x27;s all in sperm, most like by Derek talk dysfunction. So we&#x27;re gonna use caution rule. And that uses negative a sign. Next time&#x27;s a plus B co signer X minus B plus, eh? No sign of eggs. Classmates to be of sine x. You got it. Bye. It must be co signer Pegs squared. No, you can&#x27;t do right by point s. Now let&#x27;s take this guy. And don&#x27;t forget, this is Wilson&#x27;s spirit. So Let&#x27;s take this one and write it and numerator And then let&#x27;s get rid of Spirit Route. Since we have harbored do right here, we can&#x27;t let that one has negative off a plus B Co sign of X. You ride it by spurred Oh, a squared. So I&#x27;m expanding these two terms base Where plus two A B co signer fights WASPy Spirit Co signed spirit of X minus Be spread minds to a B no sign of X minus a spread. Co signed X Now to a because I&#x27;m effects, these terms will cancel out off For this term. We have negative A spread scientifics minus a B sign. Eggs have Sinus plus B squared sine X plus a be co sign X Inside X, divided by eight plus be co sign X squared now a B science coastline. Next terms for kids out. Also, we can see by this We have papers. People said it&#x27;s here. We have skirt got here so we&#x27;ll have a party for sonnets and dino man right here so we can burn the right time as signings Times base worried. Less peace word divided by oh, we have screwed off base. Birdman&#x27;s peace word waas co sign squared effects planes. Uh, this is minus based graybeards be scraped here. That is incense bird demanded by skirt A plus B comes on X Now we have a scrape My mind&#x27;s be strict term here in here so we can turn it out Let&#x27;s write the this one on a science Eggs times ace grade money He&#x27;s where? Divided by scribe Oh, a scribe Minutes be spread Well, Spike five spirit good off your mind school science for have you know that one? Let&#x27;s go Sense spirit of ecstasy with science Cardiff, Lex and screwed up Science part of X will be sign of eggs and side effects is working out We have been almost not for that this term. We also have one or a speaker sign of banks. Now we have a scribe minds be spirit Here in the numerator You also have spirit of backing denominated So we can just let out a little time skirted off a spread months be skirting the numerator What I&#x27;m the answer will be described it off a spirit friend speech Berg united by a plus. He co signed X

Amrita Bhasin · Answer

this question asked us to find the derivative of inverse tan of co sign of data. We know that we can differentiate the function with respect Tax one over one plus co sign if data squared using the derivative the derivative of coastline data is negative. Sign of there. This simplifies to negative sign of that on the talk over one plus Cho Sun data squared. This is our derivative.

Ernest Castorena · Answer

Pauline function more grandiose. View General will have 1/1. Plus the inputs quit. I only have the most played by the derivative of the input, but this one is just negative. Sign of data Looking good. So far, no, we have negative sign of data over what&#x27;s actually we have our triggered anti coastline squared plus sign of squared is equal to one is actually we can get this in terms. Uh, when we substitute coastline squared, we can get one sine squared So we have one plus one minus sine squared of data which leaves us with negative saying if they don&#x27;t over to my same square deflated Which helps us keep it all on the same But this looks good.

Victoria Dollar · Answer

Okay, so we&#x27;re asked to find the derivative of why? Well, whenever y equals a sign of X plus B co sign of eggs all over a sign of X minus B co sign of X. So what I&#x27;m gonna end up doing is I&#x27;m gonna separate these out into effigy where f is going to be a sign X plus B coastline X and my G is gonna be a sign X minus PICO sine x Because I can do f over G. If I find the derivative of this, I can use my derivative formula Where have the derivative of f times G minus the derivative of G times F all over g squared. So let&#x27;s hear on the side go ahead and find my derivative of F So I&#x27;m gonna break these down because of the derivative rules, I can actually do a sine X and then do be co sign X separately. So a times sign X so that is gonna carry down the derivative of Sine X is co sign X and bring down maybe the derivative of co sign of X is negative. Sign of X. So my answer is gonna be a co sign of X minus B sign of X because I can pull that negative in front of the beat and then repeating this with my G derivative. Again, a derivative of Sine X is co sign eggs minus B. And then again, my derivative of coastline is negative. Sign. Thanks. So I&#x27;m gonna have a co sign of X and then because I have a double negative, it becomes a positive. And just to make things simple, I&#x27;m actually gonna break down this formula even more so, What I&#x27;m going to do first is I am going to do F prime times G, and then we will put it together at the very yet. So my derivative is going to be a co sign X minus b sign of X times my G function, which is a sine x minus B co sign of X and let&#x27;s go ahead and won&#x27;t supply this up. So a times my A is going to be a squared sine x co sign X and then a times might be is gonna be minus 80 co sign at Squared X. Then I&#x27;m gonna repeat. So might be a sign of x times My A sign of X It&#x27;s gonna be minus a b sine squared x and then my my to bees I&#x27;m gonna multiply together to get plus b squared sine x co sign X Okay, so now Minnick over here and do my G prime times f So my G prime is a co sign X plus b sine of x times my f which is a sine x plus b co sign of X So I&#x27;m gonna do my two A&#x27;s together Multiply them so it&#x27;s gonna be a squared sine x co sign X Gonna do my a co sign Times might be co sign to get a B co signed squared X The multiply might be signing my a sign together So plus a b science squared X and then multiply my to piece together So plus b squared sine x co sign ICS Okay, so now when I met up doing something according to the formula almost subtract my f prime g and my G prime f So it&#x27;s going to be a squared sine x co sign X minus a p coast sine squared x minus A p sine squared x plus b squared sine X Co sign X and then I&#x27;m going to subtract my a squared. And if you notice I lined them up with those that were similar terms And since I am subtracting them, what I&#x27;m going to do is I&#x27;m gonna go through ants, put a a negative sign in each one of in front of each one to make the math a little bit easier so thes will end up cancelling out So my a squared sine x coastline Next wills will cancel because we have a plus and a minus so positive and negative the two a b co sine squared are going to both stay the same because they&#x27;re both negative. Same thing with our science corpse. However r sine X and co sign X with r b squared are going to cancel. So if I go back to my f over g formula, you remember the difference which is this here? We&#x27;re gonna combine these so my my two babies are not gonna be negative to a b co sine squared X. And then my two sine squared is minus two ap sine squared X all over my G squared, which is going to be a sine x minus B go sign X squared. And yes, this could be our final answer. However, we can simplify this down just a little bit. So if we notice in our numerator we have a negative to a B that I can pull out from each of these, which then leaves a coast sine squared eggs plus a sine squared eggs. If you remember from the trick functions coast sine squared X plus one plus signs cortex is going to give us one. So then our final answer B negative to AP over a sine x minus b Co sign X Square.

Heather Zimmers · Answer

we&#x27;re going to find the derivative of this function, and I&#x27;m going to start by changing the notation just a little bit. I&#x27;m going to change the square root notation two exponents rotation. So we have Why equals co sign of the sign of tangent Pi X to the 1/2 power. Okay, so how many layers of chain do we have here? We have the inside most layer pi times x the next layer tangent the next layer sign the next layer 1/2 power and then the outermost layer co signs. We have five altogether. So we&#x27;re going to be multiplying five derivatives together. So why Prime is the derivative of co sign the outermost one is negative sign. So have negative sign of everything that was inside it. I&#x27;m gonna go ahead and write it as a square root just for the sake of writing this so I don&#x27;t have to change it later. Okay, Now we move on and multiply that by the derivative of the next layer. And the next layer was the 1/2 power. So we bring down the 1/2 and we raised the inside to the negative 1/2 the inside is sign of Tangent of Pi X So that&#x27;s to the negative 1/2 power. Now we move to the next layer which is sign and the derivative of Sinus co sign. So we have co sign of Tangent of Pi X. Now we move to the next layer, which is tangent and the derivative of tangent is he can&#x27;t squared So we have c can&#x27;t squared of Pi X and then we moved to the innermost layer Pi x and it&#x27;s derivative would be pie. Now let&#x27;s see what we can do to simplify our answer. We have something with the negative exponents so we could bring that to the denominator. We can bring the two to the denominator from the 1/2 and we just have a single negative sign We can combine the negative sign and the pi toe lead us off So we have the opposite of pie times the sign of the square root of sign of tangent Pi X So we&#x27;ve taken care of the negative and the sign and the pie So far Now we&#x27;re bringing the 22 the bottom from the 1/2 and then since we have a negative 1/2 power here. We&#x27;ll bring that to the bottom and will make that a square root. Okay, so now we have the co sign of tangent Pi X on the top. And then we have the c can&#x27;t squared of Pi X also on the top and nothing else combines. So that&#x27;s it. That&#x27;s our lovely answer.

Problem

Find the derivative. Simplify where possible. $$ …

Problem 29 Easy Difficulty

Find the derivative. Simplify where possible.
$$
h(x)=\ln (\cosh x)
$$

Answer

$\tanh x$

Related Courses

Essential Calculus Early Transcendentals

Related Topics

Discussion

Top Calculus 1 / AB Educators

Anna Marie V.

Caleb E.

Kristen K.

Samuel H.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 3

Video Transcript

James Stewart

Essential Calculus Early Transcendentals

Anna Marie V.

Caleb E.

Kristen K.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$\tanh x$

Essential Calculus Early Transcendentals

Anna Marie V.

Caleb E.

Kristen K.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartEssential Calculus Early Transcendentals

Anna Marie V.

Caleb E.

Kristen K.

Samuel H.

James Stewart

Essential Calculus Early Transcendentals