Caitlin’s Tips on How to Make an A in Math Class

Returning to college after a 4-year “break” was probably the best thing I could have done for my GPA. When I first went to college, I was young, cocky, and cared more about chasing boys and having fun than I did about actually learning. I was burned out on school, and I had no idea what I wanted to do professionally. So I explored my options. I left college (after injuring but not debilitating my GPA) and had several jobs. I tried everything from a 911 operator, a character performer at Disney World, to eventually a stable job as a Personal Trainer. And I am still a Personal Trainer. I love my job, but I don’t want to do this for eternity. I want more. So I decided to go back to school.

Coming back to school I have noticed a lot of my fellow students making the same mistakes I made my first time. Staying up late and getting up early, partying hard on the weekends, etc can really hurt your grades, because you aren’t creating a conducive learning environment for yourself. I realize this may go on deaf ears, but I am hoping that I can share some of the wisdom I had to learn the hard way so that others don’t end up having to go back to school later and waste precious time and money. I’ll get off my soap box now and just deal them out to you.

How to Make an A in Math Class:

#1. Get enough sleep. 3 hours of sleep is not enough. I’m not just talking about the night before the exam, I’m talking EVERY night. Your brain sucks on too little sleep, and the nights of not enough sleep add up to a sleep deficit. If you don’t sleep well for more than a week, it’s almost as bad as coming to class drunk. You can’t learn this way.

#2. Do Your Homework. Homework is an easy way to raise your grade, it also happens to be assigned for a reason. If you cheat on your homework, you are cheating yourself out of learning, and you are not going to make a good grade on the exam. Do the homework, take the time it requires to make sure you understand the material. It will pay off.

#3. Keep your Cell Phone OUT OF SITE and ON SILENT/VIBRATE. Texting in class = not listening in class. Our Professor actually mentioned this last class and it’s so true. Even if all you are is distracted for a few seconds to answer a quick text, it disrupts your learning time and you won’t be nearly as focused on the lesson. It’s an hour class, just wait till you get out of it. I can’t even keep mine in my line of vision because once I see the thing vibrate I’m distracted SIMPLY because I want to know who texted me and what they want… and that distraction can keep you from getting an A.

#4. Learn to Enjoy Learning. Complaining about school and whining about studying is annoying. We all have to do it. Anyone who gets anywhere in their career has had to do it. If you spend your time hating learning, you won’t learn. You have to want to learn. If you’re in a major, you better enjoy learning about it, or you are going to HATE your job. Learning is a beautiful thing! Knowledge is power. Allow yourself to enjoy learning new things and force yourself to get excited about it! Calculus will help us science majors do well in our future classes, so we might as well enjoy it now while it’s still relatively simple.

#5. Sit closer to the Professor. Studies have proven that sitting closer to the front of the classroom improve your grade. This is probably because of many factors. First, you are right up in the action, you’ve got a great view of everything on the board or overhead projector, and you won’t have any giant heads in your way. Second, you are not going to whip out your phone or start talking to your buddy if your teacher is eyeballing you at the front of the classroom, and being so close makes you much less likely to risk being called out by the professor.

#6. Study, don’t cram. Just force yourself to do this. And do it early. Don’t try to cram. In Math studying can be more difficult because it’s like… what do I study? I know the formulas and I can do the math… so what else is there? Maybe you think you know the formulas but it can’t hurt to write them out and make sure they’re memorized (flashcards are useful for this). Go back and do old homework problems. If you feel a bit shaky on a particular area, LEARN IT and don’t stop studying it until you can write a blog post about it without having to look at any notes of any kind.

#7. Don’t be afraid to get HELP. If you aren’t getting something, and you feel completely frustrated, it’s time to seek help. Go to your professor, or a student in class who is doing well. Go online and watch tutorials. Hire a tutor. Whatever you do, don’t give up. You paid big money for this class, a little extra on a tutor is worth the price. Most departments will actually assign you a tutor for FREE if you go speak to an adviser and ask.

#8. Back off on the Partying. If you want to do well, keep your partying limited to 1 night a week (Saturday nights are good for this). You don’t want to show up hung over in class, you won’t learn. You don’t want to skip class, you won’t learn. You need a clear head to learn this stuff. Alcohol brain kills grades. Parties during the week are ESPECIALLY destructive to your learning process… so my advice? Keep it to once a week or less.I know... an old lame math joke

#9. Get Exercise. Yes, I’m a trainer, so I’m biased on this issue. However, I know I learn better when my mind and body are healthy. When I exercise, I feel happier (endorphins), I look better, my stress levels go down, etc.. Infinite reasons to get a good jog or weight training session in before studying. PS: studies have proven you learn better after working out and it has been suggested to work out in the morning before class.

#10. Eat Regularly, and Healthy. It’s a well-known fact that breakfast can make or break your test scores. It’s been studied and proven. You need to eat before you take a test. There is much debate on WHAT you should eat, but I know that a balanced and healthy diet helps. Living on vending machine food (I did this my first year of college) is horrible for your health. Feeling sick and gross all the time is horrible for your learning. Try to get in at least 3 meals per day, and if possible, 6 small meals. Keeping your blood sugar levels normal is VITAL to your ability to process and retain information, so you want to have a diet rich in protein and fats (which stabilize blood sugar). Carbohydrates are VITAL to brain health, so you need to have at least moderate amounts of carbohydrates in your diet as well, especially in the morning. It’s important to keep food additives and unnatural foods out of your diet (i.e. foods that will last through the apocalypse (*ahem*twinkies*ahem*) are probably not too healthy for us). 

#11. Don’t Skip Class. Unless you are on your death-bed, don’t skip class. Skipping class means missing important information that is SO MUCH EASIER to learn from a professor than trying to read the notes someone else took or catching up with the book (in my humble opinion, that is). I never skip class, not ever. And that is all I did my first round of college, and I got the C’s to prove it.

#12. Remove the Drama from your life. There will always be drama. However, if you have too much drama going on, and you can identify the source of the drama, it may be time to extract that source from your life. If you have a friend or boyfriend that keeps screwing up your ability to concentrate in class because they’re starting drama, or they don’t respect the time you need to devote to homework, you may just need to make a little space. I mean, if someone can’t support you chasing your dreams and goals, why are you wasting time with them? Surround yourself with people who make you a better person.
someecards.com - You're the best thing I ever found on Craigslist.
I’m not trying to be preachy here, I just know that these are what help me, and I want to help you. Happy weekend to you all!

Back from Spring Break, Test in Hand

Well, we got our tests back today and I did better than I thought I did, but missed questions I thought I’d gotten right and got questions right I thought I had done wrong. That’s kind of frustrating. So today will be dedicated to my biggest mistake, the one I corrected last time and yet STILL have not learned from. I STILL cannot find the stupid equation of the line that is tangent to the function. So I am going to put the problem up here, take myself through it step-by-step, and hopefully make it stick this time.

Find the equation of the line that is tangent to the function f(x)=x^2+6x-1 at the point (1,6).

1. First, you are going to want to use point-slope form, something I continually forget the equation for. y-y_1=m(x-x_1)

2. Then you want to solve for m by finding the derivative of F and plugging in the x-coordinate.

f(x)=x^2+6x-1 f'(x)=2x+6 if x=1, 2(1)+6 = 8 so m=8

3. y-6=8(x-1) = y=8x-8+6 = y=8x-2

It’s not hard. I just goofed. I have no idea how I got this wrong, but I managed to do so, and I don’t plan on letting myself mess it up again.

 

I hope you all had a good test score, and to sweeten my post, I wanted to add a little chart I made from the notes we did in class today:

Functions Tells us what about F When Positive When Negative When zero
F (original Function) y-coordinate Graph is above x-axis in either quadrant I or II Graph is below x-axis in either quadrant III or IV On the x-axis (x-intercepts/roots)
F’ (first derivative) -slope of the tangent line

-instantaneous rate of change

-limit of the difference quotient

-whether graph of F is increasing or decreasing

Graph of F is increasing Graph of F is decreasing -critical points

-might be a relative extremum

-horizontal tangent line

F’’ (2nd derivative) Concavity of F -The graph of F is concave up

– F’ is increasing

– The graph of F is concave down

– F’ is decreasing

– Might be inflection point (check that it actually changes concavity)

Eureka! I’ve got it! (Logs, Natural Logs, and e)

I finally got it. I had to go back to MathXL and do the optional homework on logs and natural logs, but it finally started making sense.  I want to post the log rules I hadn’t found, but desperately need to know:

1. ln(xy) = lnx + lny

Example: ln8 + lny = ln(8y)

2. ln(\frac{x}{y})=ln(x)-ln(y)

Example: ln4-2ln4+ln8=ln(\dfrac{4*8}{4^2}) which simplifies to ln(2)
3. ln(\frac{1}{x})=-lnx
4. ln(x^b)=blnx
Example: \frac{1}{2}lnx = lnx^{\frac{1}{2}}
5. log_bx can be re-written as b^y=x
Example: log_{22}x=-3 can be written as 22^{-3}=x
6. e^{lnx+lny}=xy because e^{lnx} = x
7. b^{xy}=(b^x)^y
8. b^{x+y}=b^x*b^y
9. Using those rules for e – e^{lnx} = x so e^{12lnx}=(e^{lnx})^{12}=x^{12}
10. Another example where you can use those rules with e: e^{lnx^3+3lnz}=x^3z^3
I also was having a bit of a hard time dealing with exponential x’s with our 2nd derivatives and found this to be useful:
IF you end up with a problem that looks like this: log_{81}27=x can be translated into 81^x=27 how do we solve for the exponent?  Factor the 81 and 27 like so: (3^4)^x=3^3 then solve for the exponents 4x=3 x=\frac{3}{4}
All these should be useful in helping us with second derivatives and remembering natural log rules, hopefully.
I’m excited to be having a study group with the girls I sit next to in Calculus! I think it will be really useful to hash out all the details, and hopefully do really well on Exam #2 so we can enjoy our Spring Break!
Finally, I leave you with this:
I thought this would be the most hilarious gift to give a gamer who wants a new controller.

Derivatives within Derivatives

Today as our professor went over second and third derivatives, it all finally clicked. Especially now that I know that calculus was a means to further Isaac Newton’s study in the area of Physics, I realize why derivatives help with that.

Here’s the analogy our professor gave us:

Imagine you’re driving down the highway. If the original function gives the position of the car, the first derivative gives the car’s velocity as it moves down the highway, the second derivative gives the acceleration of the car, and the third derivative is the jerk that occurs as it accelerates. Pretty interesting. So I made a visual diagram to sort of drive it home. Pun TOTALLY intended.

Note here that I’m driving a very awesome convertible. The velocity is demonstrated by my hair blowing in the wind, the acceleration is more imagined here, just imagine me with my foot pressing on the gas… “the jerk” has to do with that jerk you feel as you accelerate, and I like to think the original function here would be the dot on the GPS. If you can’t see it, just click on the photo and it will enlarge for you.

The biggest issue I’m having now is remembering all the rules for e^x and natural logs. So I’m going to be doing some more research into that and hopefully posting some very important rules asap.

History Lesson

I’ve been desperate to apply Calculus to anything else in life, anything at all. After all the point of this blog is to connect the dots, increase our understanding, etc. All week I have been trying to come up with something genius to post on my blog.

Then it hit me. When I started this class, I wanted to know WHO came up with calculus… and why? I’m about to give you all a nice history lesson so sit down, get yourself a nice cup of tea, and enjoy…

Who invented Calculus?

You have our famous friend, Sir Isaac Newton

The man responsible for all things difficult in life

and his pal Gottfried Leibniz

The Man with a lot of hair

to thank for the development of Calculus. But they by no means did it on their own… many mathematicians before them paved the way, including Archimedes, who was the first to tangent a curve.

When?

According to wikipedia, it developed throughout the 17th century after centuries of mathematical exploration that lead to it. Also, Isaac Newton got most of his breakthroughs during the plague.  I guess hard times can bring about some creative yet hard math.

Why?

Newton was getting into physics and geometry, and needed calculus style math to further develop in these two areas of study, especially physics. Leibniz was studying math and the study of the metaphysical. His goal was to create “a general method in which all truths of the reason would be reduced to a kind of calculation.” I’m not sure if calculus does this, but it sounds like a nice idea.

My Take On It

I’m hoping that since Calculus is the math that helps us in Physics, that when I take Physics this summer, I will be better prepared to kick some physics… Arse… since by then I’ll be a calculus genius.

~FIN~

Logarithmic Functions (Rules) and Derivatives of Logarithmic Functions (More Rules)

8 Basic Properties of Logarithmic Functions:

1. log_b1=0

2. log_bb=1

3. log_bb^x=x

4. b^{log_bx}=x, x>0

5. log_bMN=log_bM+log_bN

6. log_b\frac{m}{n}=log_bM-log_bN

7. log_bM^p=plog_bM

8. log_bM=log_bN if and only if M=N

7 More Rules/Quick Equations for Derivatives of Logs and Natural Logs:

1. D_x(e^x)=e^x so basically the derivative of e^x is e^x (examples to come)

2. b^x=e^{xlnb} (for example y=5^x is also equal to y=e^{xln5} which you can then solve for using the chain rule) OR you can use the following equation instead:

3. For a function y=b^x, \dfrac{dy}{dx}=b^x*lnb

4. e^{lnx}=x as well as ln(e^x)=x (this is not a derivative equation but is helpful in finding the derivative).

5. D_xlnx=\frac{1}{x} the derivative of lnx is always \frac{1}{x}

6. Change of Base Formula (also not a derivative equation but helpful) log_ba=\dfrac{log_ca}{log_cb} aka log_ba=\dfrac{lna}{lnb} for any c>0. and c≠1.

7. For any valid log base b (meaning b>0, and b≠1): when y=log_bx, \frac{dy}{dx}=\dfrac{1}{xlnb}

Examples using these equations in order:

A.Using D_x(e^x)=e^x I can show you how to use this information with the previous rules (chain, quotient, product, constant multiple) to get the derivative using e.

1. Using D_x(e^x)=e^x with the constant multiple rule:

y=7e^x

y'=7e^x using the constant multiple rule

2. using D_x(e^x)=e^x with the product rule:

y=xe^x

y'=xe^x+e^x

or in real numbers

y=x^2e^x

y'=x^2*e^x+e^x*2x

y'=x^2e^x+2xe^x

y'=e^xx(x+2) simplified.

4. using D_x(e^x)=e^x with the quotient rule:

y=\frac{x}{e^x}

y'=\dfrac{e^x*1-x*e^x}{(e^x)^2}

y'=\dfrac{e^x-xe^x}{(e^x)^2}

y'=\dfrac{1-x}{e^x} simplified.

5. using D_x(e^x)=e^x with the chain rule:

g(x)=e^{9x}

g'(x)=e^{9x}*9

g'(x)=9e^{9x}

B. Using Rule #3: For a function y=b^x, \dfrac{dy}{dx}=b^x*lnb here is an example:

y=11^{x^5+9}

y'=11^{x^5+9}*ln11

C. Using the rule \frac{d}{dx}lnx=\frac{1}{x} we can use the following example:

y=ln(2x+5)

y'=\dfrac{1}{2x+5}*2 (remember we also have to incorporate the chain rule since it’s a function within a function)

y'=\dfrac{2}{2x+5}

If you want to get a little more complicated with it, and remember some of the logarithmic rules, you can solve for even more difficult equations. Let’s take the rule lnM^p=plnM for our first example:

y=ln(x^5)

y=5lnx using that rule above

y'=5*\frac{1}{x}

y'=frac{5}{x}

Another useful rule we went over ln\frac{M}{N}=lnM-lnN

y=ln(\dfrac{\sqrt{x}}{8x^2-3})

y=ln\sqrt{x}-ln(8x^2-3)

y=lnx^{\frac{1}{2}}-ln(8x^2-3)

y=\frac{1}{2}lnx-ln(8x^2-3)

y'=\frac{1}{2x}-\frac{1}{8x^2-3}*16x

y'=\frac{1}{2x}-\frac{16x}{8x^2-3}

TADA!!!!

Finally, I’ll show you how you can use the rule lnMN=lnM+lnN

Example:

y=ln(x^4(3x+7)^8)

y=lnx^4+ln(3x+7)^8

y=4lnx+8ln(3x+7)

y'=\frac{4}{x}+\dfrac{8}{3x+7}

D. Finally I’ll use the rule y=logb_bx y'=\dfrac{1}{xlnb}

Example:

y=4log_6x

y'=\dfrac{4}{xln6}

Hopefully this was helpful to you. I want to make a video soon, but I have not yet found a decent way to work the camera while doing math. I’m working on that. I found today’s homework challenging. The last few problems took me a few tries, but I think these rules really help and if I had taken the time to bust them out for the homework I may have gotten finished faster. Meh. Next time.

Hope you all have a fantastic weekend! I found this on Pinterest and thought it was amusing.

Exam 1 – My Grade and What I Learned

Today we got our first Exam back. I did pretty well, I got an 87. This was not the A I was hoping for, but I plan on making it an A with the extra credit assignment we were given by the professor, where we look at what questions we missed on the exam, redo them the right way, with the right answer, explain what we did wrong, etc. (it’s pretty in-depth), and then show her in a private interview and hopefully get 5 extra points added to the exam grade.

 

I think I can handle it, because I only missed 3 multiple choice questions (one I totally called a few posts ago), and part of one of the open answer questions.

 

I really love this class. I thought I was going to hate it. I love the way it’s structured, I love the blog, but what I love most about this class is that our teacher is actually more concerned that we learn the material than she is about being hard on us.  What I mean by this, is that most professors make you do the homework, and you don’t have the opportunity to check your work. They grade you, and half the time you never learn from your mistakes, because that would require taking your already graded homework, looking it over, finding the answer somehow, and then doing it again the right way but with no reward (other than learning the material for a test).  Instead of doing that, our Professor has taken every single step of the way to actually ensure we learn. Homework is set up so that if we get the problem wrong, we can work a problem that is almost identical but with different numbers, and try again. It also has a built in tutorial on mathxl.com where you are shown exactly how to solve the problem. That has been a GOD send. Plus, the blog helps solidify info, apply what we’re learning to other things, and learn from classmates who are doing blogs as well. And now, even though we finished Exam 1, she’s allowing us to LEARN from our mistakes, and be credited for doing so. I think this is a very innovative, forward-thinking technique, and I assume this is why I heard from several fellow classmates last semester that I needed to take the Professor I’m learning from. I’m really not trying to suck-up here, I feel that many calculus (and other subjects) teachers/professors could learn a lot from my professor’s style. I couldn’t be happier.

Today we spent most of class going over the weekend homework, which I felt was relatively easy. Next class we’re moving into Logs and Cosine and Sine… things I recall vaguely, but have no idea what I used to do with them… I know, it’s scary when you think “I used to work with Logs and Sines and Cosines, but now I have no idea what they even ARE.” I have to give myself a little slack, though, it’s been almost a decade.

Once I get a little refresher on these things, I plan on actually making a video explaining them. If not I will dedicate a post to them probably tomorrow once I go through the homework and look at some videos on some of the linked sites I have on here.

I hope my fellow classmates did well on the test. I’ll leave you with this little gem.

 

You know Math’s taking over your life when…

You pass a sign on the highway that looks like this:

What it really said

But what I saw was this:

What I saw

I’m not kidding… I’ve started solving math problems in my head while sitting in traffic on my way home from school.

It’s not right.

F(x) = \dfrac{2x}{18}*\dfrac{2x}{20} can be simplified to F(x) = \dfrac{x^2}{90}

F'(x) = \frac{x}{45}

F'(4) = \frac{4}{45}

F'(11) = \frac{11}{45}

See??? I’m sick!  I need help!

In better news, I want to thank Calculus for helping me get a 99 on my first Chemistry test… it was so much math and I couldn’t have done it without you, Calculus class.

 

 

Feelings on Exam 1, and the newest rules for Derivatives

Exam 1 went well. Oh… except for the fact that I am 100% positive I missed the first question. All I remember was I was supposed to find the limit of x at 3, from the left, and I forgot that these little symbols

|x or whatever equation happens to be inside| mean only for positive numbers, so when you get the answer -2, the real answer should have been just 2.

I can’t remember some of the simpler rules from algebra and my previous math courses, and that really annoys me. I literally realized it the minute I walked out of the test, but it’s not like you can run back in and demand a redo.

So now I have to bank that knowledge away and never let it escape me again.

On to what we learned in class today…

The Chain Rule:

If G(x) = F( U(x) ) then G'(x) = F’ (U(x)) * U’

That’s a lot of confusing letters, so I’ll break it down for you further..

If G(x) = (3x + 5)^8 then G'(x) = 8(3x+5)^7 * 3

If you’re still confused I will show you what I did

G(x) = F( U(x) ) = G(x) = (3x+5)^8

Now we have to figure out what F is and what U(x) is.

F is x^8 and U(x) fills in for what would’ve been x in F(x). So instead of x, we put U(x) which in this case is (3x + 5)

Now that we have our two functions we can derive F(x) into F'(x) =8x^7

and U'(x) = 3

Thus G'(x) = F'(U(x))*U’ = G'(x) = 8(3x+5)^7*3 and simplified further becomes G'(x) = 24(3x+5)^7

We did some practice equations using the chain rule in class today and I found there were a few people in class who were TOTALLY confused behind me on some of the more difficult problems, so I thought I would put in 2 of the harder equations and show how I solved for them.

Using the Chain Rule with a Square Root:

Take the equation y=\sqrt{x^2+3x+1} find \frac{dy}{dx}

So to make this a little more broken down, first I am going to identify which parts of the equation I need to work with.

Let’s start with finding our “F(x)” – in this case it would be \sqrt{x}

In order to find the derivative, we have to make the square root into an exponential equation, so F'(x) = \frac{1}{2}x^{\frac{-1}{2}} If you don’t know how I got that, look at some of my older posts on square roots and derivatives.

Now we need to figure out what our “U(x)” is, in this case it’s (x^2+3x+1)

U'(x) is 2x+3 so now we can plug this in to our formula and we get this:

\frac{dy}{dx}= \frac{1}{2}(x^2+3x+1)^{\frac{-1}{2}}*(2x+3)

but we have to simplify and get it back in square root form, so because the exponent of U(x) is negative, we’ve got to put it on the bottom, and the solution looks like this:

\dfrac{dy}{dx}=\dfrac{2x+3}{2 \sqrt{x^2+3x+1}}

Easy!

Using the Chain rule with a Fractional Function:

Then I came to a fraction equation that looked like this:

s(t) = \frac{8}{t^3+1}

Momentarily my brain froze.

However, after a few minutes… the answer came to me!

What? I’m trying to keep this entertaining.

In order to do this I had to create a negative exponent, so now my equation looks like this:

s(t) = 8(t^3+1)^{-1}

Now you can easily pick out F(x) and U(x)

F(x) = 8x^{-1}

F'(x) = -8x^{-2}

U(x) = (t^3+1)

U'(x) = 3t^2

Now you can plug these into the equation and you get this:

s'(t) = -8(t^3+1)^{-2}*3t^2

s'(t) = -24t^2(t^3+1)^{-2}

and you have to put it back in fractional form so the final answer is:

s'(t) = \dfrac{-24t^2}{(t^3+1}^2

Using the Chain Rule with the Product Rule:

Finally we came to a difficult equation, and the teacher had to walk us through using the product rule with the chain rule.  Here’s what it looked like:

y=x^3(x^2+1)^5

In order to get to the chain rule, we have to use the product rule, y'=ab'+ba'

so here, we have to find a, and b, and the derivatives of both.

a=x^3

a'=3x^2

b= (x^2+1)^5

To find the derivative of b, we have to use the chain rule. So f(x) = x^5 and U(x) = x^2+1 So F'(x)=5x^4 and U'(x) = 2x Plug it in and you get:

b'= 5(x^2+1)^4 * 2x

Now we can plug in both to make the equation y’=ab’+ba’

y'= x^3*10x(x^2+1)^4+(x^2+1)^5*3x^2

This can be simplified into y'= 10x^4(x^2+1)^4+3x^2(x^2+1)^5

and further simplified by factoring out somethings.. we have

x^2(x^2+1)^4 which can be factored out and that leaves us with:

10x^2 + 3(x^2+1) = 10x^2+3x^2+3 = 13x^2+3 which we can now recombine with what we factored out for a more cohesive equation:

y'=x^2(x^2+1)^4(13x^2+3)

I hope this helps.. if it doesn’t, please feel free to ask me questions. I tried to make it comprehensible but it’s hard when the teacher is so clearly comprehensible to me so I am not always sure how to make it more so.

Finally, I solved the last equation on the front page of our worksheet, which looked scary, but it wasn’t so bad once we incorporate all the stuff we’ve learned.

Using the Chain Rule with Quotient Rule

y=\dfrac{(x^3+5)^7}{8x-3}

First what I want to do is turn this into a multiplication equation and solve for the product rule, however I wanted to try this using the quotient rule. Here goes nothing.

Quotient rule: y'= \dfrac{bt'-tb'}{b^2}

b=8x-3 and b’=8

t=(x^3+5)^7 in order to get t’ we have to use the chain rule

t'= 7(x^3+5)^6*2x^2

Now that we have everything, we can plug it in to the quotient rule:

y'= \dfrac{(8x-8)*21x^2(x^3+5)^6-7(x^3+5)^7(8)}{(8x-3)^2}

That’s a big one.

you can factor some stuff out though… (x^3+5)^6 and when you do that you can simplify what’s remaining (8x-3)(21x^3)-56(x^3+5) into 168x^3-63x^2-56x^3-280

which then becomes 112x^3-63x^2-280 you can factor a 7 out of this to get 7(16x^3-9x^2-40) and then put everything back in its place.

So now you’ve got this:

y'= \dfrac{7(x^3+5)^6(16x^3-9x^2-40)}{(8x-3)^2}

Exhausting.

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