❖ The Chain Rule : Harder Example ❖

📚 Advanced Derivatives: Applying the Quotient Rule with Multiple Chain Rule Layers 🧮

In this video, we tackle a challenging derivative problem that requires using the Quotient Rule along with the Chain Rule multiple times. This complex example is perfect for anyone looking to sharpen their calculus skills and master how to differentiate intricate functions. We'll guide you step-by-step, showing you how to apply both rules in tandem and simplify the final result using algebra.

🚀 What you’ll learn:

The formulas for both the Quotient Rule and the Chain Rule, and how to use them together effectively.
How to handle problems where you need to apply the Chain Rule multiple times within the same function.
Strategies for simplifying derivatives with algebra to get a cleaner, more concise solution.
Tips to avoid common pitfalls when working with complex functions involving both quotient and chain rules.
Whether you're preparing for AP Calculus or higher-level courses like Calculus III, this video will help you confidently solve tougher problems. By mastering how to combine these powerful derivative techniques, you’ll be able to tackle even the most difficult calculus questions.

For more advanced calculus tutorials, be sure to check out my other videos on topics like implicit differentiation, product rule, and more.

👍 Don’t forget to like, subscribe, and hit the notification bell to stay updated with my latest math tutorials! 🔔✨

Watch now to master the Quotient Rule and multiple Chain Rule applications and take your calculus skills to the next level!

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[Trademen]

I really have to say, I've been in university for a little over two months now, and my math was going to be the death of me, but your videos have really helped. I really can't explain the relief of knowing that there is somewhere I can go to get a real human being to explain something to me rather then a 5 paragraph text book definition of what chain rule is and how to apply it.

[carlsonmoncrief]

Thank you for showing the algebra involved to simplify the problem after the calculus. A lot of videos don't show simplifying the problem to the final answer, and we don't go over any of that in class since we've technically already learned it, but it's a nice refresher.

[patrickjmt]

@MIT2004 no, i did take precal in high school, but i was a pretty terrible student at that point, so i retook precal in college. my algebra has always been solid, but the trig i did not get until college.

[MrCaRTom]

dude, i just spend 2:30 hours watching your videos. crazy but honestly, you are driving me into math. thanks so much.

[ussnimitz]

Just donated $5 ! Don't stop what you're doing, you've helped me so much!

[cocowang]

You save my life !!! My math assignment due tonight and I didn't know nothing! haha!!

[patrickjmt]

@707Cookiez no problem, happy to help: )

[Abaadira]

Patrick! Your amazing man! Most teachers think they can teach just because they have a degree in math. NO! No one cares about your background in math if you can't explain it to your students. Thanks patrick. I certainly do appreciate your videos!

[amos]

Thanks very much for not keeping your knowledge but always sharing. The greatest gift men can give is to share their knowledge with the rest of the world.

[patrickjmt]

@patrickJMT other than it puts things in a standard form and makes some division by hand easier. there is no mathematical reason why it must be done though. ask your teacher and see what they say (i'm not sure, will most likely be the answer).

[ecaravello]

Great stuff Patrick. Thank you for all the great instructional videos, you are easily THE best virtual tutor I've come across on YouTube. Cheers.

[krokonaw]

I have Calc. exam tomorrow and you came to save my life!

[ngufest]

nice one uncle Patrick may God bless you

[patrickjmt]

@rainebf2 i'd give you full credit. just depends on the instructor. i have never been given a good reason as to why one needs to rationalize the denominator.

[smarties]

@patrickJMT My Calculus BC teacher also said that it makes it easier to divide by hand. He doesn't require rationalizing the denominator because it is just a remnant from the days when we didn't have calculators.

[patrickjmt]

@rycecorl yea... the chain rule can be tricky at first! stick with it though - it will come to you

[briandrost]

Thank you so much for this video. I was looking all over YouTube trying to find a video that used the chain rule like you just did. I needed a refresher for factoring out negative and fractional exponents. I wish I found this sooner because I've been stuck on the same type of problems for days. Thank you!!!

[MattOrrMusic]

@Gooldude3 : when finding the derivative of the inner function with the chain rule, , you square the denominator and since the original denominator was raised to a 1/2 power, squaring that would mean multiplying those powers, and 1/2 times 2 leaves the denominator raised to the power of 1. Thats why it looks that way.

[abdullahyusuf]

Im a Junior in college and I wanted to review derivatives just for the hell of it, and I realized idk how I would have gotten through calculus without JMT. You da man!

[hemangnehra]

Good old quotient rule and good old Patrickjmt's videos