What is the derivative of x^x? (Calculus 1 logarithmic differentiation)

Let's find the derivative of x^x by using logarithmic differentiation. This is a common problem in calculus 1 with the chain rule and implicit differentiation. Subscribe to @bprpcalculusbasics for more calculus tutorials.

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[vanadium]

Yay, red and blue pens! I'm mildly protan colorblind, so determining the difference between the red and black has always been a struggle for me.

[fanamatakecick]

The Base _e_ Method is definitely much more elegant

[wuyqrbt]

I like especially both of ways! Because all of them are just Math and I love the beauty of Math.
Great
Thank you teacher

[nicholasradley]

I've got an interesting third way to evaluate this derivative. First, define f(x, y)=x^y. We then compute ∂f/∂y=yx^(y-1) and ∂f/∂x=ln(x)x^y. We then sum the partial derivatives and let x=y to see that as desired!

[alexdemoura]

I like this video better: shaved hair, no long beard, it is more original. The 2017 version: he looked like Terence Tao. My favorite YT Math Teacher+Marathonist does not need to copycat anyone.

Anyway I will add both videos to my Factorial-Gamma Function-LogProduct W Lambert research. Thank you so much.

[ENTMusic-cjwt]

So I differentiated this function a lot and I've come to prefer the logarithmic differentiation, just because it feels like the result is coming together more nicely...
Anyway, I also tried this with the definition of the derivative but couldn't solve it. So how would you go about differentiating with the definition of a derivative?

[joshpark]

I was literally JUST thinking about this, omg you read my mind! You should also integrate it, it looks like it's fun to do!

[chrisrybak]

Nice remake of your video :-)
I prefer the lnY method with implicit differentiation. Seems neater.
But the thing that interests me most is that dy/dx seems to tend to negative infinity as x->0+ !

[vee__]

Before watching the vid I thought I'd try it and ended up using a whole different method to your two.

Let the lower x equal y. So it's f=y^x. Take the derivative using partial derivatives.. (I'll call partial deriv. 'p' instead of 'd') :

df/dx = (pf/px) + (pf/py).(dy/dx)

Then put x back in for every y, and it ends up with the right answer pretty nicely.

[swo.]

Both methods are good but my favourite is the Logarithmic Differentiation!

[jakeyounan]

They way that he can switch the markers in his one hand so seamlessly is beautiful

[nta]

I like them both. However, I'm slightly leaning towards the logarithmic version.

[TheModicaLiszt]

The second method is better! More intuitive! Thank you 🥰

[ayaanpatel]

now i am able to solve most of ur questions showed in ur videos in my own...thnx for that :)
btw i am in 12th class now

[Ninja]

I was never taught the second method ever. Only the log differentiation was taught to me on how to differentiate a function base to a function power

[MrAlvaroxz]

By simplicity (thinking) criteria, I prefer the first one, but the second one is easier of operate if you realize it earlier

[memuskhan]

The first method is how I'd approach it but the second method looks absolutely big brain. I had that "aha" moment when I saw e^xln(x)

[АндрейПетров-дэ]

I hoped, you will take the second derivative (because it's the second video about x^x derivative).

[littleretroship]

Is it still x^x bc the power rule makes it x•x^(x-1) and x is just x^1 and you add the exponents and it goes back to x^x???
Edit: oh

[samarth.patel]

Logarithmic implicit differentiation is the trick here.