Derivatives of inverse trig functions - arcsin (KristaKingMath)

Learn how to calculate the derivative of an inverse trig function. In this particular example, we'll calculate the derivative of arcsin, the inverse sine function.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

[kristakingmath]

Whenever you use chain rule and multiply by the derivative of the inside function, you always want to multiply by the entire term, not just the inside function. Basically, you're taking the derivative of arcsin, and result is 1/(sqrt(1-(2x+1)^2)). When you apply chain, you need to multiply by the derivative of the outside function, by the derivative of arcsin. Hope that makes a litte more sense! :)

[TurdFurgeson]

If you think about it, you always take the derivative of the inside of the function, even when it's just x. Only, when it's just x, the derivative equals one, so you just end up multiplying through by one -- which leaves the rest of the problem, essentially, as is. I find it helpful to reinforce this idea because then we don't train people to think there's something magical about a lone x inside a function. The rules still apply, the effects are just unnoticed.

Great and thorough coverage of this topic! I like that you showed your simplification step by step, even for students at this level of math.

[kristakingmath]

you're not rude, just giving feedback, which is awesome!! i'll try to add more videos about proofs so that you can see how to get to the formulas. :)

[brittany]

I wish I had found this channel earlier. My test is today. This made so much sense

[hassanm]

Wow, this channel is amazing. I could honestly say that this is the only reason why I am passing calculus. I just cant keep up with my professor or TA. This channel has taught me more than my university has and keeps it fun and engaging. Keep up the good work! Calc students depend on you!

[kristakingmath]

you could have -x(x+1) under the square root sign in your final answer instead of what i have, but i'm not sure ho much simpler it is. i totally know where you're coming from with the simplification stuff. the algebra is usually the hardest part of calculus for most people anyway! :)

[Dborgz]

Your angelic voice makes Calculus bearable!!! Thank you for the videos, I have brought my grade back up this semester!!!

[shierlymelany]

In Honors Cal 2 at my college and just wanted to say thanks for all your work and posting these vids! It's really great to come here and watch when I don't understand what happened in class (my teacher has a thick accent)

[kristakingmath]

That's what I'm hoping for! Thanks for the comment. :)

[kristakingmath]

Awesome! Sorry about your teacher, but I'm so happy that I can help along the way! :)

[KhmerH]

can you do another one for inverse cscx or secx? plz i'm not sure what the it means when the book says it varies base on the definitions that are used.

[mattjones]

can/ could you factor out an X in the denominator leaving 1/ x qrt of -x-1 ? jut curious cuz my professor never clearly implies if we need to simplify that far or not. or am i going to far into this? i swear i have more trouble knowing where to stop vs how to start lol

[Dannyisamazinglycool]

Thanks a bunch - final in 4 days and I've been zoned out for the last three months.

[DGFXable]

You are a gem... thank you for all your hard work.

[TheRox]

At 2:31 when u found the derivative of the function inside the brackets. You multiplied 2 by then entire fraction and not just by the inside function. I don't understand why

[sarcasm]

Ud make an awesome prof. I learn more from you than the classroom

[ifaisalme]

You are much better than my teacher thanks for the help

[Ensign_Cthulhu]

The answer puts limits on the value of X that might be worth a discussion. Specifically -x^2 - x must be > 0 for all real x and y. If I recall inequalities correctly, this means x^2 + x < 0 always, and x^2 < -x. This forces x to always be negative because -x must be positive and fractional for real x so that x^2 is always smaller. Which makes sense given that sines and cosines are always < or = to 1. (It also tells us that the tangent to the arcsine curve is vertical when x = 0.)

[ukidding]

can the derivative of inverse sine be used in any form of engineering?

[GFJR]

Calculus is done in one or two steps. What makes calculus hard is the college algebra part. Most students walk into a calculus one class with very little algebra background and so they struggle in calculus. I would like to see video clips in terms of finding the limit of functions using the delta/epsilon definition.