Related Rates #1 Problem Using Implicit Differentiation

Related Rates in Calculus: The Two Ship Problem
In this calculus video we have two ships with one initially being 100 km due west of the other. One travels north at rate of 25 kph while the other travels south at a rate of 35 kph. We want to know the rate at which the distance between them is changing some time later.
What You’ll Learn:
Understanding Related Rates: Discover the principles behind related rates problems in calculus.
Step-by-Step Approach: I’ll guide you through the process, which includes:
Creating a Diagram: Visualizing the scenario helps clarify relationships between variables.
Labeling Rates: Identify and label all rates involved in the problem.
Finding the Equation: Establish the equation that relates your variables.
Taking the Derivative: Differentiate with respect to time (d/dt) to find the relationship between the rates.
Substituting Values: Plug in specific information to solve the problem.
Why Watch This Video?
Ideal for Students: Perfect for high school and college students studying calculus and related rates.
Clear Explanations: Easy-to-follow instructions that simplify complex calculus concepts.
Enhance Your Problem-Solving Skills: Build confidence in tackling related rates problems in calculus.
📈 Don’t Forget to:
LIKE this video if you find it helpful!
SHARE with classmates or friends who want to master related rates!
SUBSCRIBE for more calculus tutorials, problem-solving techniques, and educational content!
Timestamps: 0:00 Introduction to Related Rates 1:30 Overview of the Baseball Diamond Problem 2:15 Creating the Diagram 3:00 Labeling Rates 4:30 Finding the Equation 5:45 Taking the Derivative 6:30 Substituting Values 7:15 Conclusion and Key Takeaways
#RelatedRates #Calculus #TwoShipProblem #MathTutorial #EducationalContent #LearningCalculus #ProblemSolving #HighSchoolMath #CollegeCalculus #Differential Calculus #VisualLearning #MathematicalConcepts

[@patrickjmt]

Hi all! Wanna help a KZbin education OG? Please post comments, questions and anything else on your mind in the comment section! so, don’t forget to LIKE, THUMBS UP, and SUBSCRIBE! I’d appreciate it greatly as it helps me :)

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When I watch these videos I wonder why I waste time going to class.

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I DRED my exam tomorrow... but a little less after this! Thanks

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did you pass?

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Passed the class with an A! Phew!

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how did you do?

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I'm hoping for a C but could be worse or better.

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TheAlmightyNivs keep me updated. I'm curious.

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sqielyr 52% lol

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This video's more than 10 years old and it still provides better information than my teacher.

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One thing I really love about maths is that the concepts don't change Even after a long time. I'm watching this video 12 years later and it's refreshing my mind.thank you for making everything looks so easy to do

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Never learned D.R.E.D.S., but I'm now glad that I know this before going into my Cal I test in 30 minutes. Thanks!

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we should have our campus send the tuition to you rather than to our teacher :)

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I'm so glad you're left-handed; so much easier to see what you're writing.

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this is the same textbook that i used in grade 12 calculus!!

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Thanks Patrick for all your help over the past couple years, your hard work has made a serious difference to my capability. Glad to hear you are still kicking out the tutorials, there is still so much of your material to explore!

[@djm003]

Ok im in the library at OSU and not only did this help immensely but it was word for word the problem that I was working on in my book. Great explanation. Thanks a lot!

[@msdwizzle55]

thank you so much. I have a test tomorrow on implicit differentiation and related rates. this helped me out so much.

[@mage1413]

holy crap Patrick, no matter what year of math I have taken, you always made videos on every section in my textbook. I am in first year uni now, and we're doing related rates, so this video was very helpful. Also when I was grade 10, I remember you doing a "completing the square" video which helped me out a ton. you are great

[@CanYouPleaseCall911]

I have been asking everyone what is a good way to remember the steps on related rates problems! DREDS is perfect. Now I can do the whole problem without feeling lost. I'll tell everyone that acronym. Thanks so much!

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Thank you sit! Dreds is a lifesaver on these

[@blakeburns7937]

Thank you for providing these related rates videos. They are the clearest I've seen.

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I despise related rates. You make them less despicable.

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Sarah W despicable me

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Less DREDful?

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I just might pass my calculus test....thank you!!!

[@BBrian48]

This is the EXACT same problem im doing in my book thanks. It makes sense now

[@TheLonelyNinja13]

It's amazing how you can explain something that would usually take about an hour to explain in about 10 minutes.

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You're amazing! My brain was hurting after doing this exact problem. You summarized it for me in less than 10 mins. I thank you for that sir.

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THAT'S THE EXACT PROBLEM!! MIND BLOWN..and extremely grateful

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tnx to your videos i really improve a lot in my related rates problem by watching your video you added another in my tool kit tnx

[@Nile505]

I have been watching my teacher do these for 2 classes now...and you actually showed problem solving steps..I hate my professor. She just implies that we know EVERYTHING up to this point. She never recalls anything for us, and she just skips over steps. Thanks for the vids man

[@bean2594]

When you do the trick with flipping the triangle to make one large right triangle, you are looking at the system as if you are on boat A. The side of the triangle x+y represents the relative distance between the ships as a function of time.

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I literally did this EXACT problem 3 days ago and was so confused...looked at the solution and it didn't help a great deal either. You did an awesome job explaining the process, thank you!

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[@3x10tothe8th]

Thanks for posting! I thought that I had understood this concept in class, but I guess I forgot it between then and the time I started my homework, so seeing this done again helped A LOT! Thanks again! :D

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Thanks a lot!

[@patrickjmt]

my pleasure!

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patrickJMT 6 years later and youre still helping people pass Calculus classes. Bravo.

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you've seriously helped me through this semester. thank you

[@Mike01010011]

Thank you so much. I have been struggling to understand related rates and this video really helped me understand! Thank you so much!

[@patrickjmt]

no problem. good luck with the rest of the semester

[@samisitemanmusic]

This is seriously the most helpful video on the internet!!! Thanks Patrick!!

[@patrickjmt]

@kwarkn no. you should not think about positive and negative as on an x-y plane. if the lengths are increasing as time progresses, the derivative with respect to time should be positive (to indicate it is increasing)

[@patrickjmt]

glad you like them... but everyone makes mistakes, me for sure! however, i can either delete the video or at least make annotations if it is a 'small' mistake! : )

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why does this make more sense than any lecture I ave ever been in before

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WOW, YOU SAVED MY LIFE. I've been stumped on a similar question for the past few hours now... and now that I've watched your video, I finally understand it!!!! You're way better than my calc teacher at school x___x .. thanks for sharing your knowledge!! =D

[@hondashadowfrontcaliperrebuild]

We've been studying related rates in my calc. class and this problem seemed very familiar - I found this exact problem in my book! This video was made in 2008 and my calc book was made in 2018. 10 years later they're still using the same material from old books but selling it with a new price tag.

[@patrickjmt]

welcome to over priced textbooks

[@patrickjmt]

no problem! happy to help

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OMG this is crazy! I take on online calc course and the practice problems and the tutorial aren't working but this is the EXACT problem I needed help on.

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Awesome videos. It's a nice review for my test coming up.

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The math teacher we never knew we needed

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You did a great job explaining; I understand it. Thank you.

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Thanks to you, I just might have a chance on passing my calculus exam.

[@patrickjmt]

glad i am helping! make sure and still do problems though!

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[@patrickjmt]

@durrthock that excuse does not work on me. i took 18 credit hours in college while working 40 hours a week and still found time to study the stuff i needed to study. better to say: it was not that important to me.

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[@patrickjmt]

exactly!

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you did an amazing job breaking this down

[@discorabbit]

Ah this makes review so easy. I took calculus about 4 years ago XD.

[@arooobine]

One tip that has helped me with the implicit differentiation aspect is: Always use the chain rule with individual variables. It can get pretty confusing trying to keep track of which is the dependent variable, which are independent, etc. But if you always use the chain rule (meaning the derivative of x is dx/dt etc.) you can't go wrong, because if you do so on the independent, the derivative is just 1 anyways. Just a tip that may help some people.

[@patrickjmt]

@BeBopDeluxe85 glad you like my stuff :)

[@tkeo]

Thanks! Helped me a bunch, and you explained it very clearly.

[@TuuflessTaru]

Another method which may be worth mentioning (that doesn't use implicit differentiation) is to note that the height of your right triangle, x+y, increases at 60km/h, or can be expressed as just 60t. Using Pythagoras the same way, you get z = [ (100)^2 + (60t)^2 ] ^0.5 Differentiate with respect to t, substitute t=4, and you're done! This method should be easier, since you had to use some form of z(t) in the end anyway. I suspect you were forced to diff. implicitly. :) Good job on the video!

[@mrdemonpants125]

nice example patrick, it helps a lot...thanks!

[@kimuyendinh]

Approximately 55.385 km/h

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That's what I got! Yay!!!

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Calc 1 final tomorrow... wish me luck and thanks for all your help with my homework :)

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[@patrickjmt]

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I have the same calculus book!(: I've watched your videos since last year, you're awesome! You probably get that a lot but honestly, THANKS!

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[@qwertyaznman123]

Yes, I understand that they were opposite directions, but I was merely commenting on how it might be confusing to a viewer that usually denotes x and y to perpendicular axes, and might instead use a and b rather than x and y to denote that sort of thing.

[@MRGilD0]

Thank you for the vids man its a big help I should really take calculus more serious next term hopefully this'll get me through the term anyways thanks man big help :)

[@OZZl3]

This video was extremely helpful thank you! However I find it easier to understand if you write it like this instead: z(t)^2 = 100^2 + (x(t) + y(t)^2 Dt( z(t)^2 ) = Dt( 100^2 + (x(t) + y(t))^2) (Using Chain rule, look up if you don't know it) => 2*z(t) * z'(t) = 2(x(t) + y(t)) * (x'(t) + y'(t)) Then you just juggle it around with some algebra to solve z'(t). It looks neat I think and it's easy to keep track of what the diffrent variables represent, x'(t) is the change eg km/h and x(t) etc

[@aretwodeetwo1260]

Thank you. You did a great job explaining it!

[@ian559fresno]

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[@Isk8emericas]

dreds that definitely helped man thanks

[@patrickjmt]

@petermilko it was probably the yard guy, cause my wife and i both hate vacuuming!