Calculus 1 Lecture 2.5: Finding Derivatives of Trigonometric Functions

[@AP-pm9qy]

Right now its 3 am on a Saturday and I'm staying up to watch Professor Leonard's calc vids. I told myself just one more and here I am. As someone who spends their time on Netflix and suffers from chronic laziness, Professor Leonard managed to engage me so much so that I'm binge watching his videos and taking notes like its a new and hot Netflix series that came out. You sir are legendary.

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These videos should be on netflix

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Just wish I could understand trig, But the man hynotizes me

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i can absolutely relate to this

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@@dougjohnson5487 I now am late but the prof now has a precalc section in which he covers trignometry extensively.

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You are a very mindful professor. My professor will make anybody feel like they're stupid for asking any question about algebra.

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After all these years, I have found the perfect video series. It's 6:43 am, I stayed up all night studying for an exam today (probably a bad idea), and just found these videos, and now I can only be jealous that I dont have a professor this good at teaching. Hopefully watching these lessons will save me today

[@jash8281]

did it save you ?

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@@jash8281 I dont remember about that specific exam, but I passed the class with a C so I'd say yeah it did enough

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If it’s not a secret: what university?…

[@davidsphere43]

@@mary56789 Northern Arizona University

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I can’t believe I just found this guy 6 weeks into the semester 😵 Thank you in advance for getting me through calculus!

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How'd it go?! Are you in calc III now?

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@@freezinginhere He got me an A in Calc 1! I definitely don't love math the way he does, but his lectures made it make so much more sense. I didn't even bother watching my own instructor's videos once I found this channel.

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@@absentvictim My instructor doesn't even have videos, he just posts links to Professor Leonard and The Organic Chemistry Tutor and then gives us a bunch of homework. I'm so glad I'm paying for his services -.- I wish the money went to these youtubers, since they're the ones doing all the work for my education.

[@absentvictim]

@@voxvolhynia5400 I agree completely!!

[@roblox21242]

@@voxvolhynia5400 my instruction is the same, these people should be fired they dont even try, instead uses someone's work as their own. and i looked up how much my professor makes, he makes roughly $120k a year and does nothing. imagine that.

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I wish my tuition money was going to you instead of my half-assing prof at UTA

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arlington in da house!

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Felt this in my soul 💔

[@stevensteezy8420]

9 years later and UTA still got those type of professors

[@SakibLH44]

@@stevensteezy8420 lmao that is hilarious. My god this vid gives me PTSD just thinking about those times. Good luck man.

[@Mohlawan]

9 years later. is it worth it?

[@iñigote]

Introduction and Motivation ●[0:08]. Introduction to trigonometric functions and their importance in later calculus courses. ●[0:57]. Presentation of two fundamental concepts necessary for the demonstration: ○ Trigonometric identity: sin(X) / X = 1. ○ Trigonometric identity: [1 - cos(X)] / X = 0. Demonstration of the Derivative of the Sine Function ●[2:07]. Objective: find the derivative of 𝑓(x) = sin(x), denoted as 𝑓'(x). ○ Reminder of the concept of derivative as a limit: ○ 𝑓'(x) = lim_(h → 0) [𝑓(x + h) - 𝑓(x)] / h. ●[2:55]. Application of the limit concept to the sine function: ○ 𝑓(x) = sin(x). ○ 𝑓(x + h) = sin(x + h). ○ Restriction: the expression x + h inside the sine function cannot be separated without using trigonometric identities. ●[3:32]. First step of the demonstration: ○ 𝑓'(x) = lim_(h → 0) [sin(x + h) - sin(x)] / h. ●[5:00]. Use of the angle sum trigonometric identity for sine: ○ sin(x + h) = sin(x) ⋅ cos(h) + cos(x) ⋅ sin(h). ●[6:12]. Separation of the expression into two fractions with denominator h. ○ Factoring out sin(x) in the first term. ●Algebraic manipulation to obtain the known trigonometric limits: ○ lim_(h → 0) sin(h) / h = 1. ○ lim_(h → 0) [1 - cos(h)] / h = 0. ●Application of the limits and simplification to obtain the derivative of sine: ○ 𝑓'(x) = cos(x). ●[11:15]. Interpretation: the derivative of the sine function is the cosine function, meaning that the slope of the sine curve at any point is equal to the value of the cosine at that point. Derivatives of Other Trigonometric Functions ●[12:50]. Presentation of the derivatives of the six main trigonometric functions: 1. d/dx [sin(x)] = cos(x). 2. d/dx [cos(x)] = -sin(x). 3. d/dx [tan(x)] = sec²(x). 4. d/dx [sec(x)] = sec(x)tan(x). 5. d/dx [csc(x)] = -csc(x)cot(x). 6. d/dx [cot(x)] = -csc²(x). ●[13:41]. The derivatives of tangent, secant, cosecant, and cotangent can be obtained using the quotient rule and the derivatives of sine and cosine. ●[16:17]. Observation on the relationships between the derivatives of trigonometric functions, which can help in memorizing them. Examples of Applying Trigonometric Derivatives ●[19:08]. Example 1: Find the derivative of y = x ⋅ sin(x). ○ [19:33]. Identification of the need to use the product rule. ○ Application of the product rule: ■ d/dx [x ⋅ sin(x)] = (d/dx [x] ⋅ sin(x)) + (x ⋅ d/dx [sin(x)]). ○ Simplification to obtain the derivative: ■ d/dx [x ⋅ sin(x)] = sin(x) + x ⋅ cos(x). ○ Interpretation: the obtained expression represents the slope function for the curve y = x sin(x). ●[21:35]. Example 2: Find the equation of the tangent line to the curve y = x ⋅ sin(x) at the point x = π/2. ○ Requirements for the equation of a line: a point and the slope. ○ Obtaining the slope (m) at x = π/2: ■ m = sin(π/2) + (π/2) ⋅ cos(π/2) = 1. ○ Obtaining the point (π/2, y) by substituting x = π/2 into the original equation: ■ y = (π/2) ⋅ sin(π/2) = π/2. ○ Use of the point-slope formula (y - y₁ = m(x - x₁)) to obtain the equation of the tangent line. ●[26:20]. Example 3: Find the derivative of y = sin(x) / (1 + cos(x)). ○ Identification of the need to use the quotient rule. ○ Application of the quotient rule: ■ d/dx [sin(x) / (1 + cos(x))] = {(d/dx [sin(x)] ⋅ (1 + cos(x))) - (sin(x) ⋅ d/dx [1 + cos(x)])} / (1 + cos(x))². ○ Calculation of individual derivatives and simplification. ○ [31:42]. Use of the trigonometric identity cos²(x) + sin²(x) = 1 to simplify the expression. ○ Additional simplification to obtain the derivative: ■ d/dx [sin(x) / (1 + cos(x))] = 1 / (1 + cos(x)). ●[33:40]. Example 4: Calculation of higher-order derivatives for the function sin(x). ○ Calculation of the second, third, and fourth derivatives of sin(x). ○ Observation on the cyclical nature of the derivatives of the sine function. ●[35:45]. Example 5 (verbal problem): Modeling the motion of a spring with a mass. ○ Description of the problem: a spring with a mass is stretched 3 cm from its resting position and released, oscillating without friction. ○ [41:39]. Modeling the position of the spring with a cosine function: y(t) = -3⋅cos(t). ○ Objective: find the velocity function of the spring. ○ [43:42]. Relationship between velocity and position: velocity is the derivative of position. ○ [44:04]. Calculation of the derivative of the position function: ■ y'(t) = 3 ⋅ sin(t). ○ Possible additional questions about the problem: ■ Find the points where the velocity is zero. ■ Find the equation of the tangent line at a specific point. ●[45:58]. Introduction to the chain rule. ○ Presentation of examples of derivatives that are difficult to calculate *without the chain rule*. ○ Description of the utility of the chain rule to simplify the calculation of derivatives of composite functions.

[@brianwahome2897]

I took a weekend break when you said: "We will get to the rest of it next time" at 3:53... Excellent work Prof.

[@rwharrington87]

This is fascinating after taking your Precalculus/Trigonometry course. Knowing how you can use sin and cos to model oscillations and now being able to obtain the rate of change at a given time is really enlightening. I tried going through this course several years ago, and this lecture was so perplexing at the time, not knowing the prerequisites, that I gave up. Now 4 years later it seems really easy. Filling in that missing puzzle piece in your course list was clutch sir, and needed for so many of us.

[@UndoneJigsaw]

I am preparing for my Calc class for spring 2019 with my former trig teacher in spring of 2016. I earned an A in trigonometry in her class. I got a B in Precalculus with a teacher that talked really fast. I had to take a 2 year break from school in 2017 due to life struggles but during those two years I have been watching math videos and looking over my notes. I am going to do whatever it takes to understand Calculus so I can go on towards Calc 2 and Calc 3 and Professor Leonard's videos has helped so much in understanding Math, thank you. I also noticed a pattern in the derivatives of trig functions. Every trig function that starts with a "C" (Cos, CSC, COT), the derivative is going to be a negative. Pair TAN with SEC and COT with CSC.

[@quanesshatheseventeenth8617]

Aye did you succeed bro?

[@UndoneJigsaw]

@@quanesshatheseventeenth8617 yes. I received a high A in Calculus 1, Calculus 2 and Calculus 3. I have to say thank you to professor Leonard's videos as well as taking a good instructor that was a visual learner like me. Now I have been tutoring those classes for the past 3 years.

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@@UndoneJigsaw say less. Im boutta binge these

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wow man thats awsome@@UndoneJigsaw

[@dudestaytuned4542]

I used to cry out of frustration during my first semester calculus, thank god I found this channel. Not only helped me understand but also helped my mental health. Now calc is kinda fun haha

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Can you also do workout routine??? Lol hahaha

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I wish my calculus instructor taught as well as you do. This is saving my future career. THANK YOU!

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2021...and this channel is still my favourite.

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God bless you sir, i have learnt all of my calculus from your videos so far.

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"Good deal, alright!" Haha cheers mate, your classes are brilliant, you are the reason I'll be passing calculus this year!

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For all students who visit this guy's KZbin channel"this guy deserves way more"(thanks prof for being a cool teacher I admire you...we all are lucky to find you ..)and I know I am not the only one whose complete perspective of calculus changes within one playslist You will forever be my favourite teacher❤️ Thank you my real life superman🙌 Ps(yeah I know he don't check comments)

[@michaelzanfardino5931]

that explanation at 37:25 clarified a lot... thank you... I love when questions I needed to ask that I couldn't think of are asked and answered

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[@LC-wv7tz]

Thanks for making these, like all the others have said and will continue to do. I'm a somewhat older student coming to university for the first time. At the recommendation of my college I went through a precalculus and calculus I sequence that for some reason included NO trigonometry. I didn't realize what I was getting into at the time or what I was missing out on, but I've since transferred to a new university and into a calculus II class (somehow my calc I class transferred despite the curriculum covering NO trig) and am scrambling to catch up in some ways. Of course I'm ahead in others as my old school Calculus I course covered almost all the topics concerning integration and The Fundamental Theorem normally covered in calculus II. I did take trigonometry in high school, but that was a while back. These lectures are rather helpful. All I really need is to get my hands on a good trig handbook now to cover those fundamental parts.

[@greense65]

You will generally find trigonometry covered well in just a few chapters of a precalculus book. I recommend Algebra Structure and Method by Dolciani and Brown or Advanced Mathematics by Brown. These books are written in a lean style, but given that you have done some coursework beyond this level, you should be able to handle it and will benefit in your future math by getting comfortable with this style of writing. The exercises are well-chosen. If you can do the A and B level ones, you will be fine for calculus, but I recommend working on the hard C level exercises, too, because, as I said, they are well-chosen. Dolciani also wrote Modern Advanced Analysis, also written in a lean style, which has chapters on trigonometry (and even has some other chapters introducing calculus). All these books are long out of print, but are available used at reasonable prices. I carefully worked through the first two of these books, in the order listed, and they have been a huge help. Another choice is Larson's Precalculus. This is a book still in popular use, although I would save a couple bucks and not buy the very latest edition.

[@LC-wv7tz]

Thank you. I appreciate the recommendations for textbooks and I will check them out. I was surprised, when I tried to consult my professors on this topic I was met with just some shock, they didn't have much to say on the topic of recommended texts, so I figured I'd turn to sources like this or MIT Open Courseware to do some self-study. I'm actually in school for physics and mathematics, so I will be using trig quite often and I don't feel intimidated by sitting down to read a math text. Those sound just like I was looking for.

[@greense65]

Based on your reply, I am even more comfortable recommending the Algebra Structure and Method Book II (I should have added the book II part above; there is a book I, which is first year algebra-too easy for you, and no trig) or Advanced Mathematics book to you for the trig (and precalculus in general). You seem to have more than enough mathematical maturity to handle their style, and will actually benefit from getting used to it. For other people reading this chain: In general, since you are on a calculus video website, you are most likely able to benefit from these old Brown and Dolciani books as a good precalculus reference and source for brushing up. These books can be a little tough for those intimidated by math and who don't have a good teacher. My personal story with them is that I had the Algebra I book way, way back in time (in the 8th grade). My teacher wasn't good, my parents didn't know enough math to help, and I was a pretty lazy student, so it didn't work out for me, and I actually blamed the books a lot (unfairly, instead of blaming myself-although I did blame my teacher, too, and he deserved it). As an adult, I came back to the series and really appreciated it. The writing style is an austere math writing style and there aren't tons of worked examples. But the exercise sets take you carefully from easy to pretty hard. There are solutions manuals available for sale, but in contrast to the low price of these old, used books, they are expensive.

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We need more professor like you! Too many lazy unorganized pissy teachers teaching math.

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used to hate trig in high school but first year of university this man made me love it and i feel like a genius

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18:37 LMAO. Prof Leonard makes learning calculus actually entertaining.

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Professor Leonard is like a hot buff Will Hunting

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"He's wicked smart"

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How a great explanation! Good job professor 🤓 .. your new student from Saudi Arabia 🇸🇦

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🇸🇦🇸🇦🇸🇦

[@aliozcol850]

Sir, from your previous lectures finally I've learned how to easily prove if a mathematical expression is true or not. Thank you for your work. I'm proud to say I can finally prove derivatives of trigonometric functions by myself without asking someone.

[@locky916]

you cant imagine how much your videos help use,despite 6 years have passed by. Thank you Sir. Ihope all good for you in your life

[@robertjr8205]

38:00 I am glad the student asked that question. I was thinking of that same exact thing myself throughout the video as well.

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Highlight of my life

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A great teacher👨‍🏫 Respect from INDIA

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just discovered your channel a few days ago and I'm hooked up like crazy. You are the best teacher i've ever had. Thank you!!

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you have blessed many

[@luciusirving5926]

Derivatives are awesome, easier than electromagnetism in physics. I could never do Coulomb's Law or any of those formulas related to electromagnetism like I do derivatives. Non-Newtonian physics is even more complex than linear algebra.

[@williamwelmans8648]

Professor Leonard ! You are an excellent Professor of mathematics! You have given me the slippery slope ! But thank God not the slippery slip ! Ha ha !! Thank you !!!

[@TopDedCenter]

I needed a solid refresher on calculus 1, i may end up just watching his videos through the end of calculus 3 just so i can zone out in future classes.

[@andersonmn9675]

now i remember why i used to love math...finally got a chance to come back after 50 years.

[@Mohlawan]

coming back after 10 years

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Thank you Professor Leonard

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2023 your amaizng professor ever

[@stealthworx4371]

Super grateful for this. I couldn’t for the life of me make trig work but didn’t have any issues with normal differentiation.

[@alfredndlovu193]

Prof, you are the best, excellent explanations in all of your concepts

[@sonalfernando7042]

no words to appreciate your great work sir, thank you❤

[@mrgorrila30]

11:02 "Isn't that interesting?!" "...no" "No? No. Don't care.............. You're gonna care" This guy is a savage to his students sometimes hahaha

[@yedidyakatambo2420]

very very helpful I never did calculus and now i understand everything. Its like i am actually in the classrom

[@Budams-heydays]

you are the best math teacher

[@kennyma8526]

Not fing joking. 18:02. I am watching this stuff because during covid I passed and now I am having trouble in Physics.

[@kennyma8526]

Just remember, If you want him as your professor you got to go to Merced. Merced is a city in California.

[@prim16]

38:15 The student actually had a good question... the professor's explanation helped me finally understand why d/dx(sin(x)) is cos(x). I feel so enlightened....

[@oneofyk]

Thank you, fun lecture. Enjoy being part of your (youtube) class

[@tommicheal118]

This was very helpful, Thanks

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Literally saving lives

[@bugraberkavci]

2021.. before the finals week you're saving my life

[@littlejoestar4]

damn professor leonard's shredded

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i just decided to go back to school, taking all my classes online, and your video are everything for my. THANK YOU VERY MUCH SIR

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Professor Leonard, you are amazing. I'm going to move to Idaho just so that I can audit your classes.

[@GeoRge3bood]

يستريحووون يشيخ

[@burkebecklund9041]

Thank you for making these videos.

[@sssmith41]

Lol 18:30!! Let's start selling T-Shirts to raise money for his channel!! Possibilities: "Leonard Slippery Sloped Me" "If you haven't been Slippery Sloped by Professor Leonard, You haven't lived!" "I got Slippery Sloped by Professor Leonard!!"

[@Patrick_Banks]

I started getting lost in this one

[@Therealestdmo]

Youre better than any calculus professor at UCLA lol tysm

[@khethelosiyanda9393]

i swear if you were to come to south africa we would laugh at every single one of your jokes ,we just dizzy like that😂😂(you got the quietest class ever)

[@SpaceD0C]

bc of prof. leonard i only stay up till 2am instead of 5am

[@sparkleyes1019]

11:01 - in line 3, how did we get negative sin x?

[@sourperfect]

Nilmini Deen, Applying the derivative formula for inverse sine and using the product rule for differentiating the inside function 1/x, whose derivative is -1/x^2 d/dx [arcsin(1/x)] = -1/sqrt(1-1/x^2)x^2

[@filmonhaile3327]

at 33:00 I think it can go further to 1+ SEC(X)

[@mirandaestrada628]

I feel like I am going through hyperspace going through a few of these videos.

[@Forde3654Eire]

39:13 "Super-hard word problem. Haaaarrrd!"... yep! Haha!

[@dianeallen3810]

Thank you for this!

[@hassanlaghbi7047]

Thanks our HERO!

[@elizabethvergara3436]

the thattutorguy ad is so accurate

[@zeyism3981]

Thank you Prof !!!! 💪

[@nicocapa123]

Wow 37:30 made me finally understand.

[@melissap6666]

As a note: There's a mistake at the beginning: It should read limit of x approaches 0, not h. For sin(x)/x and (1-cos(x))/x. Later the usage of h is correct. After the break both expressions are corrected.

[@cornelxbox25]

No. It's as h->0 because it comes from the difference quotient (much earlier chapter).

[@charlesamofordjuoh9940]

thank you so much for these videos

[@chemicalengineeringbd7614]

Thank you great professor

[@sarahransomm]

Outstanding, thank you.

[@cryinglemons58]

may god bless your soul

[@VarunVishwakarma1]

I made this far.. and it feels good

[@paisen4032]

in 29:04..why did the sign in between became negative?

[@nelson2k]

Damn, Prof Leonard is buff!

[@khalidbornaparte6250]

Thank you professor

[@sharanjaiswal6983]

I have one doubt at time 32:45 . Since 'x' is variable, and cos(x) could possibly be -1 somewhere, therefore 1+cos(x) could be 0 somewhere. Then, how did we cancel this expression [1+cos(x)] from numerator and denominator?

[@rapslikeaprva]

at that of x=(2n+1)pi point the function y wouldn’t exist .. so we wouldn’t be taking the derivative in the first place .. if we are taking the the derivative of a function it means that it should exist .. and so cosx is not -1.. and hence can be cancelled

[@Rahgnar2006]

Anyone else catch the reference to "Friday" at 18:17? I love it! hahahaha

[@jessicaparadis3331]

you aint got $%#& to do!! except calculus..if you want a wife lol

[@Jay-bk1fm]

This makes me miss my math classes :(

[@cate9541]

Bthis was fun to watch

[@heather0572]

Could you please turn on captions for this video?

[@singularityeh6002]

Ok I seriously need to have him as my new teacher. He has helped me to decipher Calc and help me complete my goal of completing Calc 1 by 16 years old. This is much appreciated. You have my many thanks.

[@kal5211]

Thanks again!

[@akira9195]

God bless you sir

[@peterzhao4478]

Thank you!

[@TheFarmanimalfriend]

I am having a problem with sin(h)/h and cos(h)/h as h -> 0. It seems you are dividing by zero. Can you direct me to where that proof is explained better? I am missing something. Maybe there is a textbook that explains it better? I have already taken all the Calculus classes and am going over all my classes. Calculus is, by far, the most interesting subject I have ever taken and I'm wanting to understand stuff I glossed over before.

[@akangkshyadutta2581]

kzbin.info/www/bejne/jITUgI2EirGFoLM Check it

[@JackTheGamingGuy4REALZ]

Sorry this is late, but if you've seen the earlier videos, the limit as h->0 doesn't mean h = 0, because then why use limits? We use limits to say what happens as h gets arbitrarily close to being 0, while still avoiding the domain issue. It's really a neat trick

[@kai-shanetblackwood7166]

You are saving my life.. >.

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actually useful lol good job

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good job prof

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Wish you were my teacher

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Woww, I dont know john cena is a math professor

[@rudnam]

enjoyable thank you

[@PK1857-h5f]

By the way, apart from lectures of the professor, I also like the laughter of one of the students :)

[@marie8363]

ive watched so many of these that I know who you're talking about - Scotty i think! the professor talks to him during these videos sometimes LOL

[@maxd3589]

@@marie8363 Lmao I'm here for Scott