Implicit differentiation COMPLETELY EXPLAINED! (KristaKingMath)

Implicit differentiation is the method you use to find a derivative when you can't define the original equation explicitly. Explicitly defined equations are equations that are solved for y in terms of x. In other words, equations in which the x and y variables are separated. If you can't separate the x and y variables, and you still need to take the derivative, then implicit differentiation is how you do it.

To use implicit differentiation, you can treat both variables like you would normally treat the independent variable x. In other words, whenever you come across y, you take the derivative with respect to y, and whenever you come across x, you take the derivative with respect to x. The caveat is that you always have to multiply by dy/dx, or y', whenever you take the derivative with respect to y. The reason is because you can't actually take the derivative with respect to y, since x is the independent variable. Instead, you have to take the derivative with respect to x, and since y is a function of x, when you take the derivative of y, you always have to multiply by dy/dx.

Once you've taken the derivative of both sides of the equation, you'll need to solve for dy/dx, and this will be the derivative of the original, implicitly defined 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?!”

[rkorenek]

This is the best Math Video I have seen. The clear, concise, through detail is amazing. It is this video that will convince me to take the course.

[OscarSanchez-yrrl]

WHERE HAVE YOU BEEN ALL MY LIFE KRISTA KING?! I love how you explain the math and concepts that go along with it! As a person who has struggled with math since day1, thank you!

[mimemamomumimemamo]

thank you very much, you're my savior! i finally understood it, thanks to YOU!

[StreuB]

I will be watching this video fully while reading over last thursday's lecture notes in the learning commons this afternoon. Chain rule homework is taking longer than I had hoped and its severely cut into implicit diff. HW which I should have done by class tomorrow evening. Ugh, this stuff is hard....but its so fun and rewarding at the same time.

[amyherrera]

😮 I love explanation. It really helped me understand 🎉

[Kiet]

Great lessons! I recommend this to my friend who is trying to relearn calc.

[johnkupa]

Thanks Krista King...keep up the good work.

[jamesbond_]

It's almost as if you are taking the derivatives of everything with respect to x. If you think of it that way, doesn't the dy/dx fall out naturally? Just chain rule on d(sin u)/dx = cos u du/dx; here u = y. Anyway, it's easier for me to remember it this way.

Another question: since neither x nor y "dominates" in the example you gave, I imagine you should be able to equally differentiate everything with respect to y instead of x. Except, of course, that the end goal is to yield an equation in terms of dy/dx, so I guess that biases things in favor of derivative wrt x instead of y.

Great video by the way. I didn't get implicit differentiation when I first learned of it, but thanks to you I think it will stick with me from now on :)

[chanson]

I watched the video on Khan academy but it was not sticking to my brain. It has stuck (hopefully) now : )) THANKS!

[johnstillman]

Christa (hope i have your name right I just wanted to say that without question your videos are the absolute best on youtube i thought your videos where great but the more i watch them i realize they are even better. very clear You are an amazing young lady.

[liamdsmart]

Hey great video - really helped me understand! Quick question though just to get my head around something. Would you ever use the quotient rule in implicit differentiation? Like, would you use it for a simple thing like (y/x^2)+y=100, or just make it yx^-2... :)

[maidel]

you explain really well but i still dont get why we add dy/dx whenever we have y in our equation. it doesnt make sense

[loohe]

"you can't actually take the derivative with respect to y, since x is the independent variable" - - What do you mean by that ? x and y are both dependant on each other.

[zack_]

I like your handwriting, neat and beautiful.

[paesanng]

You've only showed us how to do it, you aren't explaining why it works.