Derivative of x^(1/100) from first principles

In this video I showed how to use binomial expansion to evaluate the derivative of any power of x from first principles.

[@kingbeauregard]

Very nice! Also, I think that's my favorite hat of yours. I would follow that hat to hell and back.

[@punditgi]

Prime Newtons is awesome! 😊

[@aintgonnatakeit]

5 :)

[@joaomane4831]

Hi, teacher. Thanks for the amazing videos! I haven't tried this yet but couldn't one simply transform the exponent into a radical and then multiply by the conjugate?

[@k.kgaminganditstudy4247]

Thank you so much Sir excellent teaching.....

[@skwbusaidi]

Thank you but there is no end of the term and the last term is not h^(1/100) h is increasing by 1 each time and continue on without stop becase the exponent of x will not go to 0 so its exponent of h will never become 1/100

[@KanoShampene-wi4qt]

Thank you very much

[@PrimeNewtons]

You're welcome 😊

[@dirklutz2818]

As the power of h increases, shouldn't it be h^100 in stead of h^(1/100) for the last term?

[@johnnolen8338]

Applying the binomial theorem to a fractional exponent seems a bit hand-wavy to me. Never fear, there is still a way to salvage the proof tho. First, start by proving the chain rule from first principles. Having done that the Inverse Function Theorem follows almost by inspection. Next, if y = x^(1/100) then y^100 = x. One can easily find the derivative of y^100 from first principles using the binomial theorem. Having done that apply the Inverse Function Theorem to this result and the derivative of x^(1/100) is thus proved.

[@kingbeauregard]

This article is above my pay grade, but if I'm reading it right -- IF -- maybe not so many hands were waved. en.wikipedia.org/wiki/Binomial_coefficient#Generalization_and_connection_to_the_binomial_series

[@johnnolen8338]

@@kingbeauregard Thanks for the link. That's good information to have, but still not something I would expect beginning calculus students to know. For what it's worth I'm sure Newton's explanation was correct; it's just the why is it correct part that was unclear to me.

[@PrimeNewtons]

That makes a lot of sense

[@saharashara7980]

وشكرا استاذ طريقة ممتاز ة

[@mondayizuchukwu2733]

The beat activating

[@NanzalaJoyceMary]

thanks

[@siddhanttandon367]

how do you derive the formula for first principles?

[@johnnolen8338]

The first principles formula, aka the difference quotient is the slope of the line tangent to the curve f(x): change in the y-coordinates over the change in the x-coordinates. You then evaluate said slope as the change in the x-coordinates goes to zero. The resulting limit is the definition of the derivative. That's why this particular limit is said to be from first principles.

[@PrimeNewtons]

That's the slope of the tangent line.