Your Teacher Didn’t Show You This Hack👀

In this video, we will take the derivative of x^x. Derivatives like these can be tricky in calculus I, but after this video, you're one step closer to crushing that AP exam, ninja! We'll use the, product rule, chain rule, and logarithmic differentiation and after first rewriting the expression using a nifty hack! My hope is that the trick you learn here will help you quickly identify and solve problems like these in the future! Enjoy!
0:00 - Introduction
0:12 - First Approach (Wrong)
0:50 - Chain Rule Trap
1:37 - The Cool Trick (Log Substitution)
2:06 - Rewriting The Problem
2:27 - Chain Rule Review
3:25 - Solving Using Chain Rule
4:31 - Product Rule

[@NumberNinjaDave]

Quiz question: why can’t the answer simply be x(x^(x-1))??? Tell me below

[@CarlJohnson-mi2zo]

That e^ln(x) property is so useful for this kind of weird derivatives

[@NumberNinjaDave]

For sure

[@kiddo0142]

differentiate it as a function raised to a constant and then differentiate it as a constant raised to a function and then add the two answers together and see what you get

[@NumberNinjaDave]

Feel free to go ahead and do that to show us your work

[@kiddo0142]

@@NumberNinjaDave
d/dx(x^x) =
x(x^(x-1)) =
x(x^x)(1/x) =
x^x
d/dx(x^x)=
x^x(lnx)
1st+2nd ans=
x^x+x^xlnx

[@NumberNinjaDave]

@@kiddo0142 this method happens to get the answer but isn’t a sound approach
First 1/x needs to be taken with caution as what if x is 0?
Second, the first half didn’t really make progress. It just landed back at the original problem

[@kiddo0142]

@@NumberNinjaDave well if x=0 the original problem would be undefined as 0^0 and this is just a trick if you need to impress someone :)

[@NumberNinjaDave]

@@kiddo0142 0^0 is defined as 1 (some schools of thought say undefined, but I’m from the school of thought that there’s only one element in the power set of the empty set and so it’s an agreed on multiplicative identity). But your method allows division by 0 in the 1/x step.

[@DaanHun]

i think it would look a bit nicer to just write x^x instead of e^ln(x^x) and factor out the x^x so you get x^x(ln(x)+1). But nice video!

[@NumberNinjaDave]

Thank you! Yeah, just personal preference

[@whiteshark6004]

Great content , keep going !!!!

[@NumberNinjaDave]

I appreciate you, ninja

[@NumberNinjaDave]

What do you all think of me adding study background music in my videos? Distracting or helpful?

[@victorferreira5852]

I dont understand, what is the problem you are talking about? Is it what you just said in the video, about the power not being a constant? I dont know any other issue...

[@NumberNinjaDave]

Yes, a non constant and how to solve it