Derivative of Inverse Functions Examples & Practice Problems - Calculus

This calculus video tutorial explains how to find the derivative of an inverse function. It contains plenty of examples and practice problems for you to master the concept. it shows you how to find the inverse function f^-1(x) from f(x) and how to calculate [f^-1(x)]' in terms of x and y. It also shows you how to calculate the slope of the tangent line using the derivative. In addition, it shows you how to calculate the derivative of the inverse function using implicit differentiation - dy/dx. Examples include polynomial and square root / radical functions.
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[@TheOrganicChemistryTutor]

Derivatives - Free Formula Sheet: bit.ly/4dThzf1 Derivatives - Video Lessons: www.video-tutor.net/derivatives.html

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[@colins9616]

@ 4:39 when he says you add dy/dx onto the y derivative, that just means it's a coefficient or something you are multiplying with y. So you can divide dy/dx just like any other number. It is only used for a derivative of y. So this means if you are using a formula to find an implicit derivative, you only have to tack on dy/dx to the y's that are derived, for instance if ysquared becomes 2y. If any of the y's stay the same in the derivative, they don't need a dy/dx.

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[@AM-md4ps]

I understand this to a certain degree but I’m also lost

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I thank God everyday for your existence!

[@caroe.3237]

I really wish you would put y' so it's not confusing which is which

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i love u, u explained it better than my calc 2 teacher

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Thanks to this guy, i felt like this is the fastest 30-minute lecture i ever had

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[@hye6158]

this is literally the hardest of the derivatives... im watching this the the third time and i dont get it :(

[@niina.5190]

i feel you T-T

[@KillerBlader2118]

@@nononnomonohjghdgdshrsrhsjgd same this is the most difficult of the derivatives for me so far..

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[@bryceodom3386]

I'm pretty sure you're trying to find the tan line of the inverse function at X=2, but you said you found the tangent line of the original function, y=x^2+2x+1 , at x=2. To find the tan line of the original function you take derivative of the original function y=x^2+2x+1 and plug in 2 which gives you mTan=6 so y=6(x-2)+9 is the tangent line at x=2 of the function y=x^2+2x+1.

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I was looking exactly for this

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[@anjalichand8128]

Thank you ...for making this concept easy and for clearing my doubt

[@sanelekhumalo9314]

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[@Rickby]

Yeah this is the first video I’m really lost on, did I skip something prior to inverse functions?

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[@twelveguo5869]

Can you believe I have only a few days to learn the whole university calculus?

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Oh why?

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[@natnaelhabtamu2799]

I have nothing to say but give respect

[@Nemoguzapomnit]

Thx for your videos! Btw I've noticed that we can just find the slope for original function and then just substitute it in 1/a, where a is a slope of original function.

[@Algebrainiac]

Great vid! Although I do wonder… what if f(x) is a trig function? And you’re asked to find the derivative of the inverse? I’m pretty sure you’d have to do the inverse trig functions and then take the derivative but I’m not sure

[@Sir_oleks]

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[@yennako]

this is hella confusing 😭😭

[@El_ragel_el3nab]

Welcome to calculus

[@s8diee]

real

[@aesthete8237]

This video is great However, I got confused about this 17:32. The slope of tangent line is 1/6 using this inverse function method. But when I tried to use limits to find the slope I got 6 which is contradicting to the first answer. I just want to ask; is 1/6 the slope for the inverse function or the original function? If yes, then why it's not the same answer if you use limits to find the slope of the tangent line? Pls... notice and answer me. I'm really confused about it.

[@andrewlong7432]

This is right because 6 is the slope of your f(x) function. We know that this is correct because the slopes of a function and its inverse are reciprocals. This proves that the slope of the inverse is correctly 1/6

[@JuliuSeizure]

@@andrewlong7432when he said the question, he said find the slope of the tangent line of this function, not the inverse function

[@JuliuSeizure]

I had the same issue, i was like differentiate and substitute! Differentiate and substitute!…? What the heck are you doing??? I guess the question is wrong I’m thankful for the reply above me, the slope of a function and that if it’s inverse are reciprocal.. interesting. Sorry for the cringey comment, basically there was a problem with the question. I got 6 too. What’s this you’re saying about limits though? Differentiation from first principle?

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[@hanstamarasteinke3643]

on your third method of finding the inverse derivative when converting from square root to power and using power rule, you said (1/2-1 ) is the same as 1/2-(2/2)=1. wouldn't the correct answer be -1/2?

[@hanstamarasteinke3643]

nevermind, i watched further and see that you corrected your mistake

[@MrYogiknight]

i just looked threw the comments instantly also and I was like am I the only dumb ass and you saved the embarrassment.

[@pkus]

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[@foreignlocalmusiclyricsent2752]

Thank you always

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Great video!

[@GamingWithMicc]

You are the best man!

[@KB-WORDS]

I am trying to figure out that why it cant be that clear with teacher like think about it what the heck the prof in our school or uni trying to do, and why this guy is just making it as simple a piece of cake

[@abhimanyukumar7442]

Very nice explanation thank u sir...

[@AidenCorage]

thanks bro

[@samirmattout282]

At 12:57 when you said that x = ( y+1)^2 then y + 1 = square root of x, why can't y+1 also be - square root of x?

[@ammar3354]

I'm sooo lost

[@edimulo3903]

@@moisesinfante8470 same 😔

[@fresh_vanilla]

@@moisesinfante8470 I swear nothing is helping me understand this inverse bs

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Great 💙

[@sohailabbasnageri3659]

thank you bro for making video

[@RANDOMSTUFFROCKS1]

ngl my brain hurts

[@dx6295]

3:31 why don't need to consider -sqrt(x+7) when square root both side?

[@aliaskarbekishev8152]

the same question

[@antoniorubio602]

I dont think you can find the derivative of the inverse function for your second example because the function is not invertible right?

[@harirajagopal3979]

3 ways to find inverse function summary starts at 8:20

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Thank you sir

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You're my superhero✋🏻❤️‍🩹

[@RetroStarExploding]

In the first part where he first describes the equation, f inverse of x = y not true since its equal to y inverse?

[@easoneats]

The Goat

[@reubenwilliammpembe667]

you are the best!!!

[@andreypopov6166]

read chapter from 3 books, watched 2 videos....and still don't understand this dances with inverses :) i dont want to memorise but to understand...Anyway thank you!

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The studypug ad never fails to irritate me

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God bless u

[@amnaameir4413]

Those 27 min, I watched them in more than hour *_*

[@student-isktheyabebrahimal5377]

LMFAO

[@MysteriousMan-w8s]

1:24 I don't get it why Inversed F of F is cancelled out, can anyone explain it for me or is there a name for this?

[@Sushil-l8q]

F inverse is 1/f so 1/f×f=1

[@6dance99]

I thought I got it but then i looked at the comments and now im all fucked up again

[@babs_iq3066]

Did anyone else isolate y for the last problem just for the challenge

[@burger9997]

I love you sm

[@dudududududu12345]

just completely confused me, i used to know how to do these equations, watched the video got confused and forgot

[@outofpocket72]

Awesome video but I can’t find anything helpful on composite functions. For example, finding f’(3) and f(x) = (g(x)) type problems where g(3) = 4 and g’(3) = 7 and all that.

[@nathanhocking8073]

Well, in your example, if f(x) = g(x) and g'(3) = 7, then f'(3) =7. But in a generalized sense for finding the derivative of the inverse of f(g(x)), then first you solve for the composite function. For instance, let's say f(x) = 2x-3 and g(x) = x²-3. To solve for a composite function, you take the inner function and treat it as the x of the outer function. That would mean that f(g(x)) = 2(x²-3)-3 which simplifies to 2x²-9. We can treat this function as a new function and call it, say, h(x). So h(x) = f(g(x)) = 2x²-9. We then take the inverse of h(x), which would be x = 2h⁻¹(x)²-9. In order to isolate h⁻¹(x), we first add 9 to both sides, then divide both sides by 2, then take the square root of both sides. This leaves us with √((x+9)/2) = h⁻¹(x). We can then plug that equation into the formula taught in the video, and that will give the derivative of the inverse of f(g(x)), which in this case, would be 1/(4(√((x+9)/2))).

[@ryanmelsness3667]

I'm still confused RIP

[@wandering-willow6091]

Ok 1:34 if inverse ftn of x is equal to y and y is a ftn of x that means f inverse of x is equal to f of x? how is that even possible??😭

[@rachels2cents325]

12:57 You got the inverse wrong it's not a function because the original function don't pass the horizontal line test.

[@jameshall5171]

Rachel Garza-pinales It works for the domain of f(x) (or the range of f^-1(x)), where x >= -1. If x is less than -1 then it won’t work because you’ll get imaginary numbers.

[@nat3081]

BLESS

[@jacobfalk4827]

You didn't use trig fncns. I'm crying

[@garrettzaporowski2170]

Professor literally told me we’re not allowed to just switch the x and y how tf do I find the inverse lmao

[@胡景皓-e6s]

why the 4:51 is 2y

[@Tweatopia]

Broooo I’m failing my test I don’t get shitttt😭😭😭😪😭

[@rickyfatty363]

lmao

[@arifcalskan4933]

you king

[@newton6031]

thanks i go it

[@mamadujalloh9278]

all you have to do is take the derivative of the inverse the same way you would for any other number. No need to use the formula

[@maemillanes5357]

How are you supposed to do it when it’s just a graph w/ no given function and they want you to sketch the inverse derivative??? 😐

[@Nemoguzapomnit]

How come that x^2-7 has an inverse if it doesn't pass HLT?

[@katharinak.8731]

Because its x^2, meaning, it is a parabola and a negative value in x will always come out positive

[@Nemoguzapomnit]

@@katharinak.8731 2^2-7 is positive?

[@katharinak.8731]

@@Nemoguzapomnit No, what I am saying is that (-2)^2 is the same as (2)^2

[@Nemoguzapomnit]

@@katharinak.8731 yeah that's true. But my question was about horizontal line test

[@katharinak.8731]

@@Nemoguzapomnit right, of you draw out x^2 - 7 it is a parabola, and y values cannot occur twice for it to pass a hlt

[@aurorachee]

Dude why you know like everything from chem till cal???

[@xSabir-hc7wj]

how is y = x^2 - 7 and x = y^2 - 7, wouldn't sighs change if u re arrange it??

[@bryceodom3386]

you're not rearranging the function. All you're doing is switching X for Y and Y for X.

[@audreka13]

@@bryceodom3386 Why? Is that just a thing we have to do?

[@fletcherk2294]

Talk to fast for me to understand

[@xianmariano6422]

yup im dead

[@randallhalejr.4247]

Da derivadive

[@liamhoward2208]

Quadratic equations do not have inverses. They are not one-to-one.