What is implicit differentiation?

Most often in calculus, you deal with explicitly defined functions, which are functions that are solved for y in terms of x. In that case, finding the derivative is usually really simple, because you just call the left-side of the equation y', and then you differentiate the right side with respect to x.

But when you come across an implicitly defined function, finding the derivative isn't always that easy. Implicit functions are functions where the x and y variables are all mixed up together and can't be easily separated. That's when implicit differentiation comes in handy.

Implicit differentiation lets us take the derivative of the function without separating variables, because we're able to differentiate each variable in place, without doing any rearranging.

When we use implicit differentiation, we differentiate both x and y variables as if they were independent variables, but whenever we differentiate y, we multiply by dy/dx. That's because y, as a dependent variable, is actually a function of x. Therefore, when we take its derivative, it's as if we're taking the derivative of a composite function, and we therefore have to apply chain rule. When we do apply chain rule, we multiply by the derivative of y, which is dy/dx.

● ● ● GET EXTRA HELP ● ● ●

● ● ● CONNECT WITH KRISTA ● ● ●

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?!”

[notadvertiserfriendly]

i always had trouble with this but when you explained x^2 + y^2 = 12 and you changed y^2 to [f(x)]^2, it helped this concept make a lot more sense to me, thanks !!

[eduardocont]

FINALLY IT MAKES SENSE AND DOESN'T COME OUT OF NOWHERE. That is some sneaky thinking and I love it

[eleanorlouise]

This video is amazing, there aren't enough clearly outlined videos of calculus topics on the internet, this one really helped me!

[besthero]

THANK YOU! THIS IS A GREAT WAY OF EXPLAINING IT :D

[florentinosanchez]

Best explanation in youtube, thank you so much

[legendaryxk]

Finally understood, before i was wondering why is it implict different like just shuffle the terms around u will reach it & it turns out u can do it but its very complicated to solve organise and would have bad accuracy & impossible in some cases.

And the [f(x)] was simplest and best explaination

[taleahbailey]

This is such an awesome explanation. Thank you so much for the clarification!!

[rajpatel]

Thank you so much. I was trying to figure out "why we do implicit differentiation" and you explained it so easily. Totally got my sub.

[zack_]

👍👍👍👍👍 The BEST explanation of implicit differentiation that all other videos/URLs that I visited fail to do. Thank you very much for the great job! 3:30 forward: tell exactly why dy/dx is needed along with y as the dependent variable, crystal clear! The 2-step 'principle' of solving implicit differentiation is especially handy and intuitive. Not forgetting the enlightening statement: "implicit differentiation comes from the chain rule"! All these make this video unique and huge help for those stuck with the 'weird' implicit differentiation.

By the way, the explicit method works as well here, it is just a bit messier by involving sqrt(12-x^2) and its reciprocal, and chain rule of course. The result is the same, -x/y. So implicit diff. seems to be an alternative, easier way of solving some complex derivatives.

[kristakingmath]

Have you implicit differentiated today?! :D

[creationwt]

Wow, that helps understanding how the chain rule is involved. Thanks much. Greatly appreciated.

[keyurpatel]

U r awesome Krista ur this explanation made my day
Love u

[jagadeeshkini]

Krista! Amazing simplicity and clarity! I just loved this video (thumbs up)

[megata]

Thanks this was way easier than I thought

[qualquan]

In Implicit functions (IF) the "Dependent variable (DV) is NOT JUST a function of the Independent vaiable (IV)". So if IV is X then DV might be Y or G or a mixture of X and Y but not JUST a function of the IV or X. Some use explicit = numerical quantities and implicit for non numerical quantities but when dealing with (IF) it is more comprehensively defined as "When the DV is not just a function of IV"
Differentiation of that DV which is not a function of IV requires the chain rule.

[rshupper]

I have literally NEVER heard anyone make that connection between the x and the y variables!!!!

[moaalseiari]

4 Years does make a difference. Are you doing a PhD ? Also, I would suggest you to do a hypothesis testing videos for statistics if you got bored from all the calculus videos.

[nathan-wright-prof]

I dont quite understand, i understand in the sense of the chain rule. And i can physically solve them but i dont get mentally. If your original function is implicit and you substitute y for f(x), well thats not possible right? Because implicit means it cannot be expressed as a function without loss of detail. So how does this work in general, in this case i could assume its because a circle is symmetric.

[aelinnotalien]

You posted this video on my birthday when I turned 11.😊😊

[zalsheiba]

I didn't understand from the beginning just the definition but the way of solving I think it is different way than the professor teach us